The Diagonal Subsequence Lemma for Bounded Real Arrays

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let $a:\mathbb{N}\times\mathbb{N}\to\mathbb{R}$, $(m,k)\mapsto a_{m,k}$, be a function such that for every $k\in\mathbb{N}$ the sequence $(a_{m,k})_{m\in\mathbb{N}}$ is bounded.

(i) Diagonal subsequence. There is a strictly increasing sequence $(m_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ such that for every $k\in\mathbb{N}$ the sequence $(a_{m_{j},k})_{j\in\mathbb{N}}$ is convergent.