$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle h,e_n \rangle$

Question

I took a passage from a textbook regarding equivalent conditions of having an orthonormal sequence in a Hilbert space H. Why is the equality

$$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle h,e_n \rangle$$

true?

In the image below, I highlighted this equality in a red box.

Answer 1

Let $k\ge n$. Then by construction of $s_k$ $$ \langle s_k,e_n\rangle = \langle \sum_{i=1}^k c_ie_i,e_n\rangle = c_k, $$ where the latter equality follows from orthonormality of the $e_n$. Now $c_n = \langle h,e_n\rangle $, and the claim follows.