Uniqueness of limits and boundedness in normed spaces

Proposition. Let $(X,\|\cdot\|)$ be a normed space.

1. If a sequence $(x_n)$ converges to both $x$ and $y$, then $x=y$.
2. Every convergent sequence is bounded .

Context. These are basic structural properties of norm convergence, paralleling the corresponding facts in metric spaces.

Proof sketch.

1. Using the triangle inequality, $\|x-y\|\le \|x-x_n\|+\|x_n-y\|\to 0,$ so $\|x-y\|=0$ and hence $x=y$.
2. If $x_n\to x$, then for $n$ large we have $\|x_n-x\|\le 1$. Then $\|x_n\|\le \|x\|+1$ for all large $n$; finitely many remaining terms are bounded as well.