Define $\Vert \vec{x}\Vert_\infty = \text{max}_{1\leq i \leq n} \vert x_i\vert$. I want to prove the sequence of $n$-dimensional vectors ${\mathbf x}^{(0)}, {\mathbf x}^{(1)}, {\mathbf x}^{(2)}, \dots$ converges to ${\mathbf x}$ with respect to $\Vert \cdot \Vert_\infty$ if and only if $$ \lim_{k\rightarrow \infty} x_i^{(k)} = x_i $$ for each $i = 1, 2, \dots, n$.
I'm not really sure how to proceed in either direction, and any help would be appreciated.