What do we call a sequence with a finite limit superior? Let $(s_n)$ be a real sequence. If there exists an $\bar{s}$ such that
$$\limsup_{n \to \infty} s_n = \bar{s} < \infty,$$
what do we call the sequence? Can I say the sequence is asymptotically bounded (but the asymptotic boundedness has other meanings, e.g., a function $f(x)$ is asymptotically bounded by $x^2$)? Thanks very much!

You would call it bounded above. A sequence of real numbers $(s_n)$ has a finite $\limsup$ if and only if there is an absolute bound $M$ with $s_n \leq M$ for all $n$.
It's clear that if $s_n$ is bounded above then $\limsup s_n < \infty$. For the converse, just note that if we $\limsup s_n < \infty$ this means $s_n$ is bounded after some index $N$, and then $s_n$ is also bounded for $n < N$ since $\{s_n | n < N\}$ is a finite set.
If we want to be more formal, suppose $\limsup s_n = L < \infty$.  By definition $\limsup s_n = \lim_{N \rightarrow \infty} \sup_{n\geq N} s_n = L$, so, for any $\epsilon$ we have $K$ so that for all $N$ with $N>K$ we have $|\sup_{n\geq N} s_n - L| < \epsilon$. Taking $\epsilon = 1$, we have such a $K$, and choosing any $N>K$ we have $\sup_{n\geq N} s_n < L + 1$, and so $ s_n < L+1$ for $n \geq N$. Letting $L' = \max_{n < N} s_n$, we have $s_n \leq \max(L+1, L')$ for all $n$, and so $S_n$ is bounded above.
