The Diagonal Subsequence Lemma for Bounded Real Arrays

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, and for $s\in\mathbb{R}$ let $|s|$ be its absolute value. Let $\mathbb{N}$ be the set of natural numbers, and let $a:\mathbb{N}\times\mathbb{N}\to\mathbb{R}$ be a map on the Cartesian product, whose value at $(m,k)$ is written $a_{m,k}$. Suppose that for every $k\in\mathbb{N}$ there is a real number $R_{k}$ with $|a_{m,k}|\le R_{k}$ for every $m\in\mathbb{N}$.

Then there exists a strictly increasing sequence $(n_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ such that for every $k\in\mathbb{N}$ the sequence of real numbers $(a_{n_{j},k})_{j\in\mathbb{N}}$, a subsequence of $(a_{m,k})_{m\in\mathbb{N}}$, converges.