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Exercise :

Let ${e_1,e_2,dots, e_n}$ be an orthonormal set over the Hilbert space $H$. Show that :
$$bigg| x - sum_{k=1}^n lambda_ke_kbigg| geq bigg|x-sum_{k=1}^nlangle x,e_krangle e_k bigg|$$
for every $lambda_1,lambda_2,dots,lambda_n in mathbb R$.

Attempt :

Let $x in H$ and $varepsilon > 0$. Then, we can find $lambda_1,dots, lambda_n in mathbb R$ such that :

$$bigg| x - sum_{k=1}^n lambda_ke_kbigg| < varepsilon$$

But the element $w = x-sum_{k=1}^nlangle x,e_krangle e_k$ achieves the minimum distance of $x$ from the finite dimensional space $ F = langle e_1,dots, e_n rangle$ and then it will be

$$bigg| x-sum_{k=1}^nlangle x,e_krangle e_k bigg| leq bigg| x - sum_{k=1}^n lambda_ke_kbigg| < varepsilon$$

and thus the inequality which we were asked to show is proven.

Question : Is my approach rigorous and legit enough ? If not, any recommendations, hints or elaborations will be greatly appreciated !

Notice that $r = x - sum_{i=1}^n langle x, e_i rangle e_i$ is orthogonal to each vector $e_i$. If $lambda_1,lambda_2,ldots, lambda_n in mathbb R$, then
begin{align}
| x - sum_i lambda_i e_i |^2 &=
|x - sum_i langle x, e_i rangle e_i + sum_i langle x, e_i rangle e_i - sum_i lambda_i e_i |^2 \
&= | r + sum_i (langle x, e_i rangle - lambda_i) e_i |^2 \
&= |r|^2 + sum_i | langle x, e_i rangle - lambda_i | |e_i|^2 qquad text{(by Pythagorean theorem)}\
& geq |r|^2.
end{align}