For $\{x_n\}\in X$ do we always have $\lim_{n\to\infty}\|x_n\| = 0 \quad \text{iff}\quad\lim_{n\to \infty}x_n=0?$

Question

Let $X$ be a normed vector space. Then for $x\in X$ we have that

$$\|x\| = 0 \quad \text{if and only if} \quad x = 0.$$

Do we also have that for a sequence $\{x_n\}$ in $X$ $$\lim_{n\to \infty}\|x_n\| = 0 \quad \text{iff} \quad \lim_{n\to \infty} x_n = 0?$$

Can I take this to always be true?

Answer 1

In normed spaces in general, there is no definition of $\lim\limits_{n\to\infty} x_n = x$ except $\lim\limits_{n\to\infty} \|x_n-x\|=0.$

Answer 2

The meaning of the statement $\lim_{n\to\infty} x_n = 0$ is that for any $\varepsilon>0$, we can find an $N \in \mathbb{N}$ such that for all $n>N$, $$\lVert x_n - 0 \rVert < \varepsilon$$

The meaning of the statement $\lim_{n\to\infty} \lVert x_n \rVert = 0$ is that for any $\varepsilon>0$, we can find an $N \in \mathbb{N}$ such that for all $n>N$, $$\lvert \lVert x_n \rVert - 0 \rvert < \varepsilon$$

It should now be clear that these are equivalent.