limsup lim double sequence

Question

Very elementary, but am unable to find neither proof nor counterexample.

Let $a_{n,m}$ is be a double sequence such that for some $(\alpha_n), (\beta_n), \beta$

1. $0 < a_{n,m} \uparrow_n \alpha_m < \infty$ monotonically for every $m$.
2. $\limsup_{m \to \infty} a_{n,m} \leq \beta_n < \infty$ for every $n$.
3. $\beta_n \uparrow_n \beta < \infty$ monotonically.

Q: Does it hold that $\limsup_{m \to \infty} \alpha_m \leq \beta$ ?

This would follow if $\limsup_{m \to \infty} \lim_{n\to\infty} a_{n,m} \leq \lim_{n\to\infty} \limsup_{m\to\infty} a_{n,m}$.

Answer 1

Counterexample: $a_{n, m} = n/(n+m)$, $\alpha_m = 1$, $\beta_n=\beta=0$.