Limit of $x_n$ is unique in a pre-Hilbert space

Question

If the limit of $x_n$ exists, then it is unique in pre-Hilbert space. How can I prove that?

Answer 1

Suppose $x_n \to x,y.$ Then $$\|x-y\|=\|x-x_n+x_n-y\|\leq \|x_n-x\|+\|x_n-y\|\to 0.$$