Convergence in normed spaces

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

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

A sequence $(x_n)$ in $X$ converges to $x\in X$ (in norm) if

\|x_n-x\|\to 0 \quad \text{as } n\to\infty.

Equivalently: for every $\varepsilon>0$ there exists $N$ such that $\|x_n-x\|<\varepsilon$ for all $n\ge N$ .

Context. By the metric induced by a norm, this is exactly convergence in the associated metric space.

Examples

In $\mathbb{R}^2$ with $\|\cdot\|_2$ , the sequence $x_n=(1/n,0)$ converges to $(0,0)$ .
In $C([0,1])$ with $\|\cdot\|_\infty$ , the functions $f_n(t)=t/n$ satisfy $\|f_n-0\|_\infty=1/n\to 0$ , so $f_n\to 0$ .
A sequence may converge under one norm and not under another in infinite-dimensional spaces (choice of norm matters).