The Diagonal Subsequence Lemma for Bounded Real Arrays

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $a:\mathbb{N}\times\mathbb{N}\to\mathbb{R}$, with values $a_{m,k}=a(m,k)$, be such that for every $k\in\mathbb{N}$ the sequence $(a_{m,k})_{m\in\mathbb{N}}$ is bounded.

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