Inner product on a sequence and its limit

Question

I am stuck on a question, and it seems like I'm missing a really obvious Cauchy-Schwarz application or something, but I am left scratching my head.

Let $(x_n):n \in \mathbb{N}$ be a sequence in a Hilbert space $H$. Let $x$ satisfy $\|x_n\|\to \|x\|$ and $\langle x,x_n\rangle \to \langle x,x\rangle$. Show that $x_n \to x$.

I have found so far that $\|x_n-x\|^2=\langle x_n,x_n-x\rangle -\langle x,x_n-x\rangle $, and I know that the rightmost term tends to zero which helps, but I don't know about the first one.

Any solutions? Thanks in advance.

Answer 1

Also the left term tends to zero as it is $$ \langle x_n, x_n - x \rangle = ||x_n||^2 -\langle x_n,x \rangle \to ||x||^2 - ||x||^2 = 0 $$ where I used both the hypotheses.