Little question about Banach Space

Question

It is possible that if I have a sequence $\{x_{nm}\}_{nm}$ in a Banach Space $V$ such that

1. $x_{nm}\to_{m\to\infty}y_n$ for every $n\in \mathbb{N}$

2. $y_n\to_{n\to \infty} x$

than $x_{nn}\to x$ ?

If it is not possible, what is a contro-example?

What conditions should I consider to get this result?

Answer 1

Yes, take the sequence $(x_{nm})_{n,m\in\mathbb{N}}\in\mathbb{R}^{\mathbb{N}^2}$ defined by $$ x_{nm} = \frac{n}{m}, \qquad n,m\geq 1\,. $$ Then $$ \lim_{m\to\infty} x_{nm} = 0 $$ and of course $\lim_{n\to\infty} 0 = 0$; but $$ \lim_{m\to\infty} x_{nn} = \lim_{n\to\infty} 1 = 1 $$