Algebra of limits in normed spaces

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

1. If $x_n\to x$ and $y_n\to y$, then $x_n+y_n\to x+y$.
2. If $x_n\to x$ and $\alpha_n\to \alpha$ (in $\mathbb{R}$ or $\mathbb{C}$), then $\alpha_n x_n\to \alpha x$.

Context. These are the basic “limit laws” for sequences in normed linear settings.

Proof sketch.

1. By the triangle inequality, $\|(x_n+y_n)-(x+y)\|\le \|x_n-x\|+\|y_n-y\|\to 0.$
2. Write $\|\alpha_n x_n-\alpha x\|\le \|\alpha_n(x_n-x)\|+\|(\alpha_n-\alpha)x\| =|\alpha_n|\,\|x_n-x\|+|\alpha_n-\alpha|\,\|x\|.$ Use that $(\alpha_n)$ is bounded and both factors tend to $0$.