Source: Elementary Analysis By Kenneth A. Ross
Theorem: If $s_n$ converges to a positive real number $s$ and $(t_n)$ is any sequence, then $$\lim \sup s_nt_n = s\cdot\lim \sup t_n$$
Proof: We first show $$\lim \sup s_nt_n \ge s \cdot \lim \sup t_n \; \; \; \; \; \; \; \; \text{(1)}$$ Let $\beta = \lim \sup t_n$
(Not writing case 1)
Case 2: Suppose $\beta = +\infty$
There exists a subsequence $t_{n_k}$ of $t_n$ such that $\lim_{k \rightarrow\infty}t_{n_k} = +\infty$.
Since $\lim_{k \rightarrow \infty}s_{n_k} = s > 0$, it follows that $\lim_{k \rightarrow \infty}s_{n_k}t_{n_k} = + \infty$. Hence $\lim \sup s_nt_n = +\infty$, so (1) clearly holds.
I have two questions here -
$1.$ How did the last line reach the conclusion that $\lim \sup s_nt_n = +\infty$?
$2.$ In the inequality (1), both LHS and RHS are infinity. How are we comparing two infinities with an inequality?