Show the following Inner product problem

Question

Im stuck in this problem it seems easy but I can't find the way to show it

Show that $y \perp x_n $ and $x_n \to x$ together imply $x \perp y$

Thanks for you time.

Answer 1

You have $\langle y, x_n \rangle = 0$ for all $n$ and the function $z \mapsto \langle y, z \rangle$ is continuous, hence $\langle y, x_n \rangle \to \langle y, x \rangle$, hence $\langle y, x \rangle = 0$.

Another (essentially equivalent) way of looking at it is to let $S = \{ z | \langle y, z \rangle = 0 \}$ and notice that $S$ is closed because it is the inverse image of the closed set $\{0\}$ under the continuous map $z \mapsto \langle y, z \rangle$. Then since $x_n \in S$, $x_n \to x$ and $S$ is closed, we have $x \in S$.