Convergence

Convergence in normed spaces
A sequence $$\{\mathbf{v}_k\}_{k=1}^{\infty}$$ in a normed vector space $$V$$ converges to $$\mathbf{v} \in V$$ if(Def 2.1.5 )

$$||\mathbf{v-v}_k||\to 0$$ as $$k\to \infty$$.

Precisely, this means that for any $$\epsilon>0$$, there exists $$N\in \mathbb{N}$$ such that $$||\mathbf{v-v}_k||<\epsilon$$ for any $$k\ge N$$.

Convergence is also denoted alternatively alternatively in the following two forms

(i) $$\mathbf{v}_k\to \mathbf{v}$$ as $$k\to\infty$$

(ii) $$v=\lim_{k\to\infty}\mathbf{v}_k$$