Property of set $left {y in Bbb R : y =limlimits_{n rightarrow infty} f(x_n), text {for some sequence } x

Property of set $left {y in Bbb R : y =limlimits_{n rightarrow infty} f(x_n), text {for some sequence } x_n...

Let $f : Bbb R longrightarrow Bbb R$ be a continuous function and $A subseteq Bbb R$ be defined by

$$A=left {y in Bbb R : y =lim_{n rightarrow infty} f(x_n), text {for some sequence } x_n rightarrow +infty right }.$$

Then the set $A$ is necessarily

$(1)$ a connected set.

$(2)$ a compact set.

$(3)$ a singleton set.

$(4)$ None of the above.

How to tackle this problem? Any valuable suggestions will be highly appreciated.

Thank you in advance.

edited Dec 8 at 5:29

asked Dec 7 at 18:54

math maniac.

214

closed as off-topic by José Carlos Santos, GNUSupporter 8964民主女神地下教會, Federico, Cesareo, user10354138 Dec 8 at 6:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, GNUSupporter 8964民主女神地下教會, Federico, Cesareo, user10354138

If this question can be reworded to fit the rules in the help center, please edit the question.

1

what would you say?
– Federico
Dec 7 at 19:02

I think (1) is the correct one.
– math maniac.
Dec 7 at 19:19

1

What we have to show is that if $a,b in A$ and $c in (a,b)$ then $c in A$.
– math maniac.
Dec 7 at 19:22

1

@Federico the most trivial theorem in continuity I remember is "intermediate value theorem".
– math maniac.
Dec 7 at 19:26

1

You mean $f(z_n)to c$. Yes, that is what you have to do
– Federico
Dec 7 at 19:35

|
show 12 more comments

Let $f : Bbb R longrightarrow Bbb R$ be a continuous function and $A subseteq Bbb R$ be defined by

$$A=left {y in Bbb R : y =lim_{n rightarrow infty} f(x_n), text {for some sequence } x_n rightarrow +infty right }.$$

Then the set $A$ is necessarily

$(1)$ a connected set.

$(2)$ a compact set.

$(3)$ a singleton set.

$(4)$ None of the above.

How to tackle this problem? Any valuable suggestions will be highly appreciated.

Thank you in advance.

edited Dec 8 at 5:29

asked Dec 7 at 18:54

math maniac.

214

closed as off-topic by José Carlos Santos, GNUSupporter 8964民主女神地下教會, Federico, Cesareo, user10354138 Dec 8 at 6:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, GNUSupporter 8964民主女神地下教會, Federico, Cesareo, user10354138

If this question can be reworded to fit the rules in the help center, please edit the question.

1

what would you say?
– Federico
Dec 7 at 19:02

I think (1) is the correct one.
– math maniac.
Dec 7 at 19:19

1

What we have to show is that if $a,b in A$ and $c in (a,b)$ then $c in A$.
– math maniac.
Dec 7 at 19:22

1

@Federico the most trivial theorem in continuity I remember is "intermediate value theorem".
– math maniac.
Dec 7 at 19:26

1

You mean $f(z_n)to c$. Yes, that is what you have to do
– Federico
Dec 7 at 19:35

|
show 12 more comments

Let $f : Bbb R longrightarrow Bbb R$ be a continuous function and $A subseteq Bbb R$ be defined by

$$A=left {y in Bbb R : y =lim_{n rightarrow infty} f(x_n), text {for some sequence } x_n rightarrow +infty right }.$$

Then the set $A$ is necessarily

$(1)$ a connected set.

$(2)$ a compact set.

$(3)$ a singleton set.

$(4)$ None of the above.

How to tackle this problem? Any valuable suggestions will be highly appreciated.

Thank you in advance.

edited Dec 8 at 5:29

asked Dec 7 at 18:54

math maniac.

214

Let $f : Bbb R longrightarrow Bbb R$ be a continuous function and $A subseteq Bbb R$ be defined by

$$A=left {y in Bbb R : y =lim_{n rightarrow infty} f(x_n), text {for some sequence } x_n rightarrow +infty right }.$$

Then the set $A$ is necessarily

$(1)$ a connected set.

$(2)$ a compact set.

$(3)$ a singleton set.

$(4)$ None of the above.

How to tackle this problem? Any valuable suggestions will be highly appreciated.

Thank you in advance.

real-analysis sequences-and-series continuity compactness connectedness

edited Dec 8 at 5:29

asked Dec 7 at 18:54

math maniac.

214

edited Dec 8 at 5:29

asked Dec 7 at 18:54

math maniac.

214

edited Dec 8 at 5:29

asked Dec 7 at 18:54

math maniac.

214

asked Dec 7 at 18:54

math maniac.

214

asked Dec 7 at 18:54

math maniac.

214

closed as off-topic by José Carlos Santos, GNUSupporter 8964民主女神地下教會, Federico, Cesareo, user10354138 Dec 8 at 6:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, GNUSupporter 8964民主女神地下教會, Federico, Cesareo, user10354138

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by José Carlos Santos, GNUSupporter 8964民主女神地下教會, Federico, Cesareo, user10354138 Dec 8 at 6:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

"This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, GNUSupporter 8964民主女神地下教會, Federico, Cesareo, user10354138

If this question can be reworded to fit the rules in the help center, please edit the question.

1

what would you say?
– Federico
Dec 7 at 19:02

I think (1) is the correct one.
– math maniac.
Dec 7 at 19:19

1

What we have to show is that if $a,b in A$ and $c in (a,b)$ then $c in A$.
– math maniac.
Dec 7 at 19:22

1

@Federico the most trivial theorem in continuity I remember is "intermediate value theorem".
– math maniac.
Dec 7 at 19:26

1

You mean $f(z_n)to c$. Yes, that is what you have to do
– Federico
Dec 7 at 19:35

|
show 12 more comments

1

what would you say?
– Federico
Dec 7 at 19:02

I think (1) is the correct one.
– math maniac.
Dec 7 at 19:19

1

What we have to show is that if $a,b in A$ and $c in (a,b)$ then $c in A$.
– math maniac.
Dec 7 at 19:22

1

@Federico the most trivial theorem in continuity I remember is "intermediate value theorem".
– math maniac.
Dec 7 at 19:26

1

You mean $f(z_n)to c$. Yes, that is what you have to do
– Federico
Dec 7 at 19:35

what would you say?
– Federico
Dec 7 at 19:02

I think (1) is the correct one.
– math maniac.
Dec 7 at 19:19

What we have to show is that if $a,b in A$ and $c in (a,b)$ then $c in A$.
– math maniac.
Dec 7 at 19:22

@Federico the most trivial theorem in continuity I remember is "intermediate value theorem".
– math maniac.
Dec 7 at 19:26

You mean $f(z_n)to c$. Yes, that is what you have to do
– Federico
Dec 7 at 19:35

|
show 12 more comments

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