Prove that the sequence of functions $g_{n}in C[0,4]$
I want to show that the function defined by $g_n:[0,4]to Bbb{R}$, defined bybegin{align} g_n(t)=begin{cases}0,& text{if};0leq tleq 2,\dfrac{n}{2}(t-2),& text{if};2leq tleq 2+frac{2}{n},\1 &text{if};2+frac{2}{n}leq tleq 4.end{cases} end{align}
is continuous and Cauchy.
MY TRIAL
The sequence of functions is clearly continuous by Pasting Lemma. Next, we show that the sequence of functions is Cauchy.
So, let $m,ninBbb{N},$ be given such that $mgeq n.$ Then,
$$|g_m(t)-g_n(t)|=|0-0|+left|dfrac{m}{2}(t-2)-dfrac{n}{2}(t-2)right|left|1-1right|=0<epsilon$$
I don't know if this is right. If not, kindly help please!
real-analysis functional-analysis analysis normed-spaces
edited Dec 11 '18 at 22:54
user289143
36018
asked Dec 11 '18 at 20:57
Mike
1,506321
|
show 2 more comments
I want to show that the function defined by $g_n:[0,4]to Bbb{R}$, defined bybegin{align} g_n(t)=begin{cases}0,& text{if};0leq tleq 2,\dfrac{n}{2}(t-2),& text{if};2leq tleq 2+frac{2}{n},\1 &text{if};2+frac{2}{n}leq tleq 4.end{cases} end{align}
is continuous and Cauchy.
MY TRIAL
The sequence of functions is clearly continuous by Pasting Lemma. Next, we show that the sequence of functions is Cauchy.
So, let $m,ninBbb{N},$ be given such that $mgeq n.$ Then,
$$|g_m(t)-g_n(t)|=|0-0|+left|dfrac{m}{2}(t-2)-dfrac{n}{2}(t-2)right|left|1-1right|=0<epsilon$$
I don't know if this is right. If not, kindly help please!
real-analysis functional-analysis analysis normed-spaces
edited Dec 11 '18 at 22:54
user289143
36018
asked Dec 11 '18 at 20:57
Mike
1,506321
functions are continuous. sequences are Cauchy
– mathworker21
Dec 11 '18 at 20:59
to prove Cauchy, just calculate the max difference between $g_n$ and $g_m$. what exactly is your difficulty?
– mathworker21
Dec 11 '18 at 21:00
@mathworker21: Edited some things. Max difference?
– Mike
Dec 11 '18 at 21:06
wait, you're saying $|g_m(t)-g_n(t)| = 0$ for each $t$?
– mathworker21
Dec 11 '18 at 21:08
by max difference, what I mean is $max_t |g_m(t)-g_n(t)|$. you want to show this is less than $epsilon$ for $m,n$ large enough
– mathworker21
Dec 11 '18 at 21:08
|
show 2 more comments
I want to show that the function defined by $g_n:[0,4]to Bbb{R}$, defined bybegin{align} g_n(t)=begin{cases}0,& text{if};0leq tleq 2,\dfrac{n}{2}(t-2),& text{if};2leq tleq 2+frac{2}{n},\1 &text{if};2+frac{2}{n}leq tleq 4.end{cases} end{align}
is continuous and Cauchy.
MY TRIAL
The sequence of functions is clearly continuous by Pasting Lemma. Next, we show that the sequence of functions is Cauchy.
So, let $m,ninBbb{N},$ be given such that $mgeq n.$ Then,
$$|g_m(t)-g_n(t)|=|0-0|+left|dfrac{m}{2}(t-2)-dfrac{n}{2}(t-2)right|left|1-1right|=0<epsilon$$
I don't know if this is right. If not, kindly help please!
real-analysis functional-analysis analysis normed-spaces
edited Dec 11 '18 at 22:54
user289143
36018
asked Dec 11 '18 at 20:57
Mike
1,506321
I want to show that the function defined by $g_n:[0,4]to Bbb{R}$, defined bybegin{align} g_n(t)=begin{cases}0,& text{if};0leq tleq 2,\dfrac{n}{2}(t-2),& text{if};2leq tleq 2+frac{2}{n},\1 &text{if};2+frac{2}{n}leq tleq 4.end{cases} end{align}
is continuous and Cauchy.
MY TRIAL
The sequence of functions is clearly continuous by Pasting Lemma. Next, we show that the sequence of functions is Cauchy.
So, let $m,ninBbb{N},$ be given such that $mgeq n.$ Then,
$$|g_m(t)-g_n(t)|=|0-0|+left|dfrac{m}{2}(t-2)-dfrac{n}{2}(t-2)right|left|1-1right|=0<epsilon$$
I don't know if this is right. If not, kindly help please!
real-analysis functional-analysis analysis normed-spaces
real-analysis functional-analysis analysis normed-spaces
edited Dec 11 '18 at 22:54
user289143
36018
asked Dec 11 '18 at 20:57
Mike
1,506321
edited Dec 11 '18 at 22:54
user289143
36018
asked Dec 11 '18 at 20:57
Mike
1,506321
edited Dec 11 '18 at 22:54
user289143
36018
edited Dec 11 '18 at 22:54
user289143
36018
edited Dec 11 '18 at 22:54
user289143
36018
36018
asked Dec 11 '18 at 20:57
Mike
1,506321
asked Dec 11 '18 at 20:57
Mike
1,506321
asked Dec 11 '18 at 20:57
Mike
1,506321
1,506321
functions are continuous. sequences are Cauchy
– mathworker21
Dec 11 '18 at 20:59
to prove Cauchy, just calculate the max difference between $g_n$ and $g_m$. what exactly is your difficulty?
– mathworker21
Dec 11 '18 at 21:00
@mathworker21: Edited some things. Max difference?
– Mike
Dec 11 '18 at 21:06
wait, you're saying $|g_m(t)-g_n(t)| = 0$ for each $t$?
– mathworker21
Dec 11 '18 at 21:08
by max difference, what I mean is $max_t |g_m(t)-g_n(t)|$. you want to show this is less than $epsilon$ for $m,n$ large enough
– mathworker21
Dec 11 '18 at 21:08
|
show 2 more comments
functions are continuous. sequences are Cauchy
– mathworker21
Dec 11 '18 at 20:59
to prove Cauchy, just calculate the max difference between $g_n$ and $g_m$. what exactly is your difficulty?
– mathworker21
Dec 11 '18 at 21:00
@mathworker21: Edited some things. Max difference?
– Mike
Dec 11 '18 at 21:06
wait, you're saying $|g_m(t)-g_n(t)| = 0$ for each $t$?
– mathworker21
Dec 11 '18 at 21:08
by max difference, what I mean is $max_t |g_m(t)-g_n(t)|$. you want to show this is less than $epsilon$ for $m,n$ large enough
– mathworker21
Dec 11 '18 at 21:08
functions are continuous. sequences are Cauchy
– mathworker21
Dec 11 '18 at 20:59
functions are continuous. sequences are Cauchy
– mathworker21
Dec 11 '18 at 20:59
to prove Cauchy, just calculate the max difference between $g_n$ and $g_m$. what exactly is your difficulty?
– mathworker21
Dec 11 '18 at 21:00
to prove Cauchy, just calculate the max difference between $g_n$ and $g_m$. what exactly is your difficulty?
– mathworker21
Dec 11 '18 at 21:00
@mathworker21: Edited some things. Max difference?
– Mike
Dec 11 '18 at 21:06
@mathworker21: Edited some things. Max difference?
– Mike
Dec 11 '18 at 21:06
wait, you're saying $|g_m(t)-g_n(t)| = 0$ for each $t$?
– mathworker21
Dec 11 '18 at 21:08
wait, you're saying $|g_m(t)-g_n(t)| = 0$ for each $t$?
– mathworker21
Dec 11 '18 at 21:08
by max difference, what I mean is $max_t |g_m(t)-g_n(t)|$. you want to show this is less than $epsilon$ for $m,n$ large enough
– mathworker21
Dec 11 '18 at 21:08
by max difference, what I mean is $max_t |g_m(t)-g_n(t)|$. you want to show this is less than $epsilon$ for $m,n$ large enough
– mathworker21
Dec 11 '18 at 21:08
|
show 2 more comments
2 Answers
2
active
oldest
votes
What meaning exactly do you assign to the sentence “the sequence $g_n$ is Cauchy”?
This sequence of functions converges pointwise to the function $h$ such that $h(x)=0$ if $0 leq x leq 2$ and $h(x)=1$ otherwise.
Now $h$ is not continuous whereas every $g_n$ is. So there cannot be max-norm convergence.
Edit: Just to provide some even more explicit argument because it does not seem to be accepted that the space of continuous functions from $[0,4]$ to $mathbb{R}$ is complete for the max-norm.
$g_nleft(2+frac{2}{2n}right)=frac{n}{2}frac{2}{2n}=frac{1}{2}$, while $g_{2n}left(2+frac{2}{2n}right)=1$.
Thus the max-norm of $g_n-g_{2n}$ is not lower than $frac{1}{2}$, hence the sequence is not Cauchy.
answered Dec 11 '18 at 22:06
Mindlack
1,69717
1
C([0,4]) is complete for the max-norm.
– Mindlack
Dec 11 '18 at 22:40
1
Besides, I do not have neough reputation to comment your post, but I fail to see how you can have $(N-n)/N < 1/N$.
– Mindlack
Dec 11 '18 at 22:49
If I am not mistaken, you are thus proving that the max-norm of $|g_n-g_m|$ is lower than $(N-n)/n$, which would be negative?
– Mindlack
Dec 11 '18 at 23:25
John B: I do not have another question. I am not convinced by user289143’s proof and I cannot comment it.
– Mindlack
Dec 11 '18 at 23:30
1
User289143: then you cannot have $|N-n|<1$.
– Mindlack
Dec 11 '18 at 23:31
|
show 3 more comments
A sequence is Cauchy if for every $epsilon$ there exists $N in mathbb{N}$ such that $forall n, m > N$ we have $|g_n-g_m|< epsilon$.
Now let $m > n$. How does the function $(g_m-g_n)(t)$ look like?
begin{align}
(g_m-g_n)(t)= begin{cases}
0 & if 0 leq t leq 2
\(frac{m}{2}-frac{n}{2})(t-2) & if 2 < t leq 2+frac{2}{m}
\1-frac{n}{2}(t-2) & if 2+frac{2}{m} < t leq 2+frac{2}{n}
\ 0 & if 2+frac{2}{n} < t leq 4
end{cases}
end{align}
Now $||g_m-g_n||_1=int_2^{2+frac{2}{m}}|(frac{m}{2}-frac{n}{2})(t-2)|dt+int_{2+frac{2}{m}}^{2+frac{2}{n}}|1-frac{n}{2}(t-2)|dt$
If we compute explicitly the integrals we get $||g_m-g_n||_1=frac{m-n}{m^2}+frac{(m-n)^2}{nm^2}=frac{m-n}{mn}<frac{m}{mn}=frac{1}{n}$
Let $N in mathbb{N}$ such that $N > frac{1}{epsilon}$ and we have $||g_m-g_m||_1<frac{1}{n}<frac{1}{N}<epsilon$ and this proves the sequence is Cauchy
answered Dec 11 '18 at 21:48
user289143
36018
2
Could you please explain why |N-n| < 1?
– Mindlack
Dec 12 '18 at 0:01
Edited the answer with the $||cdot||_1$ norm, to prove the space is not complete
– user289143
Dec 12 '18 at 13:22
add a comment |
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2 Answers
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2 Answers
2
active
oldest
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oldest
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active
oldest
votes
What meaning exactly do you assign to the sentence “the sequence $g_n$ is Cauchy”?
This sequence of functions converges pointwise to the function $h$ such that $h(x)=0$ if $0 leq x leq 2$ and $h(x)=1$ otherwise.
Now $h$ is not continuous whereas every $g_n$ is. So there cannot be max-norm convergence.
Edit: Just to provide some even more explicit argument because it does not seem to be accepted that the space of continuous functions from $[0,4]$ to $mathbb{R}$ is complete for the max-norm.
$g_nleft(2+frac{2}{2n}right)=frac{n}{2}frac{2}{2n}=frac{1}{2}$, while $g_{2n}left(2+frac{2}{2n}right)=1$.
Thus the max-norm of $g_n-g_{2n}$ is not lower than $frac{1}{2}$, hence the sequence is not Cauchy.
answered Dec 11 '18 at 22:06
Mindlack
1,69717
1
C([0,4]) is complete for the max-norm.
– Mindlack
Dec 11 '18 at 22:40
1
Besides, I do not have neough reputation to comment your post, but I fail to see how you can have $(N-n)/N < 1/N$.
– Mindlack
Dec 11 '18 at 22:49
If I am not mistaken, you are thus proving that the max-norm of $|g_n-g_m|$ is lower than $(N-n)/n$, which would be negative?
– Mindlack
Dec 11 '18 at 23:25
John B: I do not have another question. I am not convinced by user289143’s proof and I cannot comment it.
– Mindlack
Dec 11 '18 at 23:30
1
User289143: then you cannot have $|N-n|<1$.
– Mindlack
Dec 11 '18 at 23:31
|
show 3 more comments
What meaning exactly do you assign to the sentence “the sequence $g_n$ is Cauchy”?
This sequence of functions converges pointwise to the function $h$ such that $h(x)=0$ if $0 leq x leq 2$ and $h(x)=1$ otherwise.
Now $h$ is not continuous whereas every $g_n$ is. So there cannot be max-norm convergence.
Edit: Just to provide some even more explicit argument because it does not seem to be accepted that the space of continuous functions from $[0,4]$ to $mathbb{R}$ is complete for the max-norm.
$g_nleft(2+frac{2}{2n}right)=frac{n}{2}frac{2}{2n}=frac{1}{2}$, while $g_{2n}left(2+frac{2}{2n}right)=1$.
Thus the max-norm of $g_n-g_{2n}$ is not lower than $frac{1}{2}$, hence the sequence is not Cauchy.
answered Dec 11 '18 at 22:06
Mindlack
1,69717
1
C([0,4]) is complete for the max-norm.
– Mindlack
Dec 11 '18 at 22:40
1
Besides, I do not have neough reputation to comment your post, but I fail to see how you can have $(N-n)/N < 1/N$.
– Mindlack
Dec 11 '18 at 22:49
If I am not mistaken, you are thus proving that the max-norm of $|g_n-g_m|$ is lower than $(N-n)/n$, which would be negative?
– Mindlack
Dec 11 '18 at 23:25
John B: I do not have another question. I am not convinced by user289143’s proof and I cannot comment it.
– Mindlack
Dec 11 '18 at 23:30
1
User289143: then you cannot have $|N-n|<1$.
– Mindlack
Dec 11 '18 at 23:31
|
show 3 more comments
What meaning exactly do you assign to the sentence “the sequence $g_n$ is Cauchy”?
This sequence of functions converges pointwise to the function $h$ such that $h(x)=0$ if $0 leq x leq 2$ and $h(x)=1$ otherwise.
Now $h$ is not continuous whereas every $g_n$ is. So there cannot be max-norm convergence.
Edit: Just to provide some even more explicit argument because it does not seem to be accepted that the space of continuous functions from $[0,4]$ to $mathbb{R}$ is complete for the max-norm.
$g_nleft(2+frac{2}{2n}right)=frac{n}{2}frac{2}{2n}=frac{1}{2}$, while $g_{2n}left(2+frac{2}{2n}right)=1$.
Thus the max-norm of $g_n-g_{2n}$ is not lower than $frac{1}{2}$, hence the sequence is not Cauchy.
answered Dec 11 '18 at 22:06
Mindlack
1,69717
What meaning exactly do you assign to the sentence “the sequence $g_n$ is Cauchy”?
This sequence of functions converges pointwise to the function $h$ such that $h(x)=0$ if $0 leq x leq 2$ and $h(x)=1$ otherwise.
Now $h$ is not continuous whereas every $g_n$ is. So there cannot be max-norm convergence.
Edit: Just to provide some even more explicit argument because it does not seem to be accepted that the space of continuous functions from $[0,4]$ to $mathbb{R}$ is complete for the max-norm.
$g_nleft(2+frac{2}{2n}right)=frac{n}{2}frac{2}{2n}=frac{1}{2}$, while $g_{2n}left(2+frac{2}{2n}right)=1$.
Thus the max-norm of $g_n-g_{2n}$ is not lower than $frac{1}{2}$, hence the sequence is not Cauchy.
answered Dec 11 '18 at 22:06
Mindlack
1,69717
edited Dec 12 '18 at 11:57
answered Dec 11 '18 at 22:06
Mindlack
1,69717
answered Dec 11 '18 at 22:06
Mindlack
1,69717
answered Dec 11 '18 at 22:06
Mindlack
1,69717
1,69717
1
C([0,4]) is complete for the max-norm.
– Mindlack
Dec 11 '18 at 22:40
1
Besides, I do not have neough reputation to comment your post, but I fail to see how you can have $(N-n)/N < 1/N$.
– Mindlack
Dec 11 '18 at 22:49
If I am not mistaken, you are thus proving that the max-norm of $|g_n-g_m|$ is lower than $(N-n)/n$, which would be negative?
– Mindlack
Dec 11 '18 at 23:25
John B: I do not have another question. I am not convinced by user289143’s proof and I cannot comment it.
– Mindlack
Dec 11 '18 at 23:30
1
User289143: then you cannot have $|N-n|<1$.
– Mindlack
Dec 11 '18 at 23:31
|
show 3 more comments
1
C([0,4]) is complete for the max-norm.
– Mindlack
Dec 11 '18 at 22:40
1
Besides, I do not have neough reputation to comment your post, but I fail to see how you can have $(N-n)/N < 1/N$.
– Mindlack
Dec 11 '18 at 22:49
If I am not mistaken, you are thus proving that the max-norm of $|g_n-g_m|$ is lower than $(N-n)/n$, which would be negative?
– Mindlack
Dec 11 '18 at 23:25
John B: I do not have another question. I am not convinced by user289143’s proof and I cannot comment it.
– Mindlack
Dec 11 '18 at 23:30
1
User289143: then you cannot have $|N-n|<1$.
– Mindlack
Dec 11 '18 at 23:31
1
1
C([0,4]) is complete for the max-norm.
– Mindlack
Dec 11 '18 at 22:40
C([0,4]) is complete for the max-norm.
– Mindlack
Dec 11 '18 at 22:40
1
1
Besides, I do not have neough reputation to comment your post, but I fail to see how you can have $(N-n)/N < 1/N$.
– Mindlack
Dec 11 '18 at 22:49
Besides, I do not have neough reputation to comment your post, but I fail to see how you can have $(N-n)/N < 1/N$.
– Mindlack
Dec 11 '18 at 22:49
If I am not mistaken, you are thus proving that the max-norm of $|g_n-g_m|$ is lower than $(N-n)/n$, which would be negative?
– Mindlack
Dec 11 '18 at 23:25
If I am not mistaken, you are thus proving that the max-norm of $|g_n-g_m|$ is lower than $(N-n)/n$, which would be negative?
– Mindlack
Dec 11 '18 at 23:25
John B: I do not have another question. I am not convinced by user289143’s proof and I cannot comment it.
– Mindlack
Dec 11 '18 at 23:30
John B: I do not have another question. I am not convinced by user289143’s proof and I cannot comment it.
– Mindlack
Dec 11 '18 at 23:30
1
1
User289143: then you cannot have $|N-n|<1$.
– Mindlack
Dec 11 '18 at 23:31
User289143: then you cannot have $|N-n|<1$.
– Mindlack
Dec 11 '18 at 23:31
|
show 3 more comments
A sequence is Cauchy if for every $epsilon$ there exists $N in mathbb{N}$ such that $forall n, m > N$ we have $|g_n-g_m|< epsilon$.
Now let $m > n$. How does the function $(g_m-g_n)(t)$ look like?
begin{align}
(g_m-g_n)(t)= begin{cases}
0 & if 0 leq t leq 2
\(frac{m}{2}-frac{n}{2})(t-2) & if 2 < t leq 2+frac{2}{m}
\1-frac{n}{2}(t-2) & if 2+frac{2}{m} < t leq 2+frac{2}{n}
\ 0 & if 2+frac{2}{n} < t leq 4
end{cases}
end{align}
Now $||g_m-g_n||_1=int_2^{2+frac{2}{m}}|(frac{m}{2}-frac{n}{2})(t-2)|dt+int_{2+frac{2}{m}}^{2+frac{2}{n}}|1-frac{n}{2}(t-2)|dt$
If we compute explicitly the integrals we get $||g_m-g_n||_1=frac{m-n}{m^2}+frac{(m-n)^2}{nm^2}=frac{m-n}{mn}<frac{m}{mn}=frac{1}{n}$
Let $N in mathbb{N}$ such that $N > frac{1}{epsilon}$ and we have $||g_m-g_m||_1<frac{1}{n}<frac{1}{N}<epsilon$ and this proves the sequence is Cauchy
answered Dec 11 '18 at 21:48
user289143
36018
2
Could you please explain why |N-n| < 1?
– Mindlack
Dec 12 '18 at 0:01
Edited the answer with the $||cdot||_1$ norm, to prove the space is not complete
– user289143
Dec 12 '18 at 13:22
add a comment |
A sequence is Cauchy if for every $epsilon$ there exists $N in mathbb{N}$ such that $forall n, m > N$ we have $|g_n-g_m|< epsilon$.
Now let $m > n$. How does the function $(g_m-g_n)(t)$ look like?
begin{align}
(g_m-g_n)(t)= begin{cases}
0 & if 0 leq t leq 2
\(frac{m}{2}-frac{n}{2})(t-2) & if 2 < t leq 2+frac{2}{m}
\1-frac{n}{2}(t-2) & if 2+frac{2}{m} < t leq 2+frac{2}{n}
\ 0 & if 2+frac{2}{n} < t leq 4
end{cases}
end{align}
Now $||g_m-g_n||_1=int_2^{2+frac{2}{m}}|(frac{m}{2}-frac{n}{2})(t-2)|dt+int_{2+frac{2}{m}}^{2+frac{2}{n}}|1-frac{n}{2}(t-2)|dt$
If we compute explicitly the integrals we get $||g_m-g_n||_1=frac{m-n}{m^2}+frac{(m-n)^2}{nm^2}=frac{m-n}{mn}<frac{m}{mn}=frac{1}{n}$
Let $N in mathbb{N}$ such that $N > frac{1}{epsilon}$ and we have $||g_m-g_m||_1<frac{1}{n}<frac{1}{N}<epsilon$ and this proves the sequence is Cauchy
answered Dec 11 '18 at 21:48
user289143
36018
2
Could you please explain why |N-n| < 1?
– Mindlack
Dec 12 '18 at 0:01
Edited the answer with the $||cdot||_1$ norm, to prove the space is not complete
– user289143
Dec 12 '18 at 13:22
add a comment |
A sequence is Cauchy if for every $epsilon$ there exists $N in mathbb{N}$ such that $forall n, m > N$ we have $|g_n-g_m|< epsilon$.
Now let $m > n$. How does the function $(g_m-g_n)(t)$ look like?
begin{align}
(g_m-g_n)(t)= begin{cases}
0 & if 0 leq t leq 2
\(frac{m}{2}-frac{n}{2})(t-2) & if 2 < t leq 2+frac{2}{m}
\1-frac{n}{2}(t-2) & if 2+frac{2}{m} < t leq 2+frac{2}{n}
\ 0 & if 2+frac{2}{n} < t leq 4
end{cases}
end{align}
Now $||g_m-g_n||_1=int_2^{2+frac{2}{m}}|(frac{m}{2}-frac{n}{2})(t-2)|dt+int_{2+frac{2}{m}}^{2+frac{2}{n}}|1-frac{n}{2}(t-2)|dt$
If we compute explicitly the integrals we get $||g_m-g_n||_1=frac{m-n}{m^2}+frac{(m-n)^2}{nm^2}=frac{m-n}{mn}<frac{m}{mn}=frac{1}{n}$
Let $N in mathbb{N}$ such that $N > frac{1}{epsilon}$ and we have $||g_m-g_m||_1<frac{1}{n}<frac{1}{N}<epsilon$ and this proves the sequence is Cauchy
answered Dec 11 '18 at 21:48
user289143
36018
A sequence is Cauchy if for every $epsilon$ there exists $N in mathbb{N}$ such that $forall n, m > N$ we have $|g_n-g_m|< epsilon$.
Now let $m > n$. How does the function $(g_m-g_n)(t)$ look like?
begin{align}
(g_m-g_n)(t)= begin{cases}
0 & if 0 leq t leq 2
\(frac{m}{2}-frac{n}{2})(t-2) & if 2 < t leq 2+frac{2}{m}
\1-frac{n}{2}(t-2) & if 2+frac{2}{m} < t leq 2+frac{2}{n}
\ 0 & if 2+frac{2}{n} < t leq 4
end{cases}
end{align}
Now $||g_m-g_n||_1=int_2^{2+frac{2}{m}}|(frac{m}{2}-frac{n}{2})(t-2)|dt+int_{2+frac{2}{m}}^{2+frac{2}{n}}|1-frac{n}{2}(t-2)|dt$
If we compute explicitly the integrals we get $||g_m-g_n||_1=frac{m-n}{m^2}+frac{(m-n)^2}{nm^2}=frac{m-n}{mn}<frac{m}{mn}=frac{1}{n}$
Let $N in mathbb{N}$ such that $N > frac{1}{epsilon}$ and we have $||g_m-g_m||_1<frac{1}{n}<frac{1}{N}<epsilon$ and this proves the sequence is Cauchy
answered Dec 11 '18 at 21:48
user289143
36018
edited Dec 12 '18 at 13:21
answered Dec 11 '18 at 21:48
user289143
36018
answered Dec 11 '18 at 21:48
user289143
36018
answered Dec 11 '18 at 21:48
user289143
36018
36018
2
Could you please explain why |N-n| < 1?
– Mindlack
Dec 12 '18 at 0:01
Edited the answer with the $||cdot||_1$ norm, to prove the space is not complete
– user289143
Dec 12 '18 at 13:22
add a comment |
2
Could you please explain why |N-n| < 1?
– Mindlack
Dec 12 '18 at 0:01
Edited the answer with the $||cdot||_1$ norm, to prove the space is not complete
– user289143
Dec 12 '18 at 13:22
2
2
Could you please explain why |N-n| < 1?
– Mindlack
Dec 12 '18 at 0:01
Could you please explain why |N-n| < 1?
– Mindlack
Dec 12 '18 at 0:01
Edited the answer with the $||cdot||_1$ norm, to prove the space is not complete
– user289143
Dec 12 '18 at 13:22
Edited the answer with the $||cdot||_1$ norm, to prove the space is not complete
– user289143
Dec 12 '18 at 13:22
add a comment |
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functions are continuous. sequences are Cauchy
– mathworker21
Dec 11 '18 at 20:59
to prove Cauchy, just calculate the max difference between $g_n$ and $g_m$. what exactly is your difficulty?
– mathworker21
Dec 11 '18 at 21:00
@mathworker21: Edited some things. Max difference?
– Mike
Dec 11 '18 at 21:06
wait, you're saying $|g_m(t)-g_n(t)| = 0$ for each $t$?
– mathworker21
Dec 11 '18 at 21:08
by max difference, what I mean is $max_t |g_m(t)-g_n(t)|$. you want to show this is less than $epsilon$ for $m,n$ large enough
– mathworker21
Dec 11 '18 at 21:08