Given
$ \limsup s_n = + \infty $ and $ \liminf t_n > 0$
To prove $$ \limsup s_nt_n = +\infty$$
There exists $s_{n_k}$ such that $ \lim s_{n_k} = +\infty $. Also we have $ \lim t_{n_k} = \liminf t_n $.
$$\lim s_{n_k}t_{n_k} = +\infty $$
Since $\limsup s_nt_n$ is largest all limits of subsequences $s_nt_n$.So $\limsup s_nt_n = +\infty $
Unsure about last part. Can you guide ? Thanks