Does convergence along a diagonal grid imply convergence along other paths going to infinity?

Question

Suppose that for each natural $k$, we have a sequence of real numbers $a_{n,k}$, and that $\lim_{n \to \infty} a_{n,k}=\alpha \in \mathbb{R}$ is independent of $k$.

Suppose furthermore that $\lim_{n \to \infty} a_{n,n}=\alpha $ as well.

Is it true that $\lim_{n \to \infty} a_{n,f(n)}=\alpha$, for any monotonic increasing function $f:\mathbb{N} \to \mathbb{N}$?

Answer 1

$$a_{n,k}= \frac{k}{n^2}$$

fixing n would definitely give you a divergent sequence while your conditions are satisfied for this double sequence.

Answer 2

Take any $f$ such that for all $n$, $f(n) \neq n$ (for example, $f(n) = n + 1)$. Define $a_{n, k} = \begin{cases} \alpha & k \neq f(n) \\ \beta & k = f(n)\end{cases}$. Check what are $\lim\limits_k a_{n, k}$, $\lim\limits_n a_{n, n}$ and $\lim\limits_n a_{n, f(n)}$.