Let $\left(a_{mn}\right)_{m \geq 1, n \geq 1}$ be a double sequence of real numbers such that\\
(a) For every $n$, $b_{n} := \lim_{m \rightarrow \infty} a_{mn}$ exists;\\
(b) For all strictly increasing sequences $\left(m_{k}\right)_{k \geq 1}$ and $\left(n_{k}\right)_{k \geq 1}$ of positive integers, $\lim_{k \rightarrow \infty} a_{m_{k}n_{k}} = 1$.
Show that the sequence $\left(b_{n}\right)_{n \geq 1}$ converges to $1$.