Prove or disprove: $ \lim _{n \to \infty} \lim_{k \to \infty} a_{k, n} = \lim_{k \to \infty} a_{k,k}$ [closed]

Let $\{a_{k,n}\}_{k\geq 1, n \geq 1}$ be a sequence of non-negative real numbers where $a_{k,n} \leq M $ for all $k \geq 1$ and $n \geq 1$. Assume that $a_{k,n}$ is increasing in $k$ and decreasing in $n$. Furthermore, assume that the limit

$$\lim_{k \to \infty} a_{k,k}$$

exists and is finite. Prove or disprove

$$ \lim _{n \to \infty} \lim_{k \to \infty} a_{k, n} = \lim_{k \to \infty} a_{k,k}$$