Is $\cos(\pi \sqrt n)u[n]$ stable?
The system with impulse response given by $h[n] = \cos(\pi\sqrt{n})u[n]$ is BIBO-unstable because the sum $\sum_{n=-\infty}^\infty |h[n]]$ diverges instead of being convergent as is needed for BIBO-stability. Note that
for all positive integers $k$, $h[k^2]=\cos(\pi k)$ has value $\pm 1$ and so
$$\sum_{n=-\infty}^\infty |h[n]] = \sum_{n=0}^\infty |\cos((\pi\sqrt{n})|$$
is a sum that contains infinitely many $+1$ terms (and all the other terms are guaranteed to be positive too since $\cos(\pi r)=0$ if and only if $r = k+\frac 12$ where $k$ is an integer, and there is no integer $n$ whose square root is of the form $k+\frac 12$). So, $\sum_{n=-\infty}^\infty |h[n]]$ diverges, and the system is BIBO-unstable.