Matrices/Convergent entries/Pointwise convergent linear mapping/Exercise

Let ${}M_{k}={\left({\left(a_{ij}\right)}_{k}\right)}_{1\leq i\leq m,1\leq j\leq n}$ be a sequence of real ${}m\times n$-matrices, and let

$\varphi _{k}\colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m}$

denote the corresponding sequence of linear mappings. Show that the sequences of the entries ${}{\left({a_{ij}}\right)}_{k}$ converge for all ${}i,j$ if and only if the sequence of the mappings converge pointwise.

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