Let $\{{e_n}\}\ n=1,2,3,...$be an orthonormal sequence in a Hilbert space $H$ and let $x\not=0 \in H$ then $<x, e_n> \to 0$ as $\ n \to \infty$.
By Bessel's inequality we have $\sum_{n=1}^{\infty}|<x,e_n>|^2\leq||x||^2$ and therefore $<x, e_n> \to 0$ as $\ n \to \infty$.
Is my argument correct? But the answer says the limit is $1$. Could you please explain to me.
Your argument is correct: since$$\sum_{n=1}^\infty\langle x,e_n\rangle^2\leqslant\lVert x\rVert^2,$$$\lim_{n\to\infty}\langle x,e_n\rangle^2=0$ and therefore $\lim_{n\to\infty}\langle x,e_n\rangle=0$. The limit could not possibly be $1$ for each $x\neq0$. Take, for instance, $x=e_1$. Then the sequence $\bigl(\langle x,e_n\rangle\bigr)_{n\in\mathbb N}$ is just $1,0,0,0,\ldots$