In Rudin, $\limsup$ is defined as follows:
Let $S$ be the set of subsequential limits of $\{s_n\}$. Then $$\limsup s_n = \sup S. \tag{1}$$
However, our real analysis instructor defined $\limsup$ in a different manner:
$$\limsup s_n = \lim_{n \to \infty} \sup_{m \ge n} s_m. \tag{2}$$
I am having trouble understanding how these two definitions are equivalent. It would be very helpful to me if somebody could provide a proof with some explanation.
My thoughts on the problem:
I have noticed that the usual trend with these sort of proofs is to prove the upper bound $(1) \le (2)$ and then the lower bound $(1) \ge (2)$ to get the desired conclusion. However, I am unsure how to even begin.