Show that a convergent orthogonal sequence in a Hilbert space has limit zero

Question

Let $(x_n)$ be an orthogonal sequence in a Hilbert Space $H$, that is $\langle x_n,x_m\rangle =0$ when $n\neq m$. Show that, if $(x_n)$ converges, the limit must be $0$.

My ideas: Every convergent sequence is also a Cauchy sequence, and therefore the result follows, considering $\langle x_n-x_m,x_n-x_m\rangle = \langle x_n,x_n\rangle + \langle x_m,x_m\rangle - \langle x_m,x_n\rangle - \langle x_n,x_m\rangle$. Is that correct?

Answer 1

I think your proof is unfinished. Suppose $x$ is the limit of the sequence $\{x_n\}$ and consider that $$ 0 = \langle x_n,x_m\rangle. $$ First let $n\to\infty$ and then let $m\to\infty$ in the above expression. What do you get?

Answer 2

Your proof could be continued this way: for $n>m$ we have $$\|x_n-x_m\|^2=\|x_n\|^2+\|x_m\|^2\ge \|x_m\|^2$$ Therefore $\|x_m\|\to 0.$