I'm currently studying self-studying elementary analysis, and my book (Lay) skims over something I'm not sure I fully get. I just want to double check my logic before I go any further.
This is what the book says: Sps. $\{s_n\}$ is a sequence bounded by some real number $S$, and $m = $ lim sup $s_n$, then no number larger han $m$ can be a subsequential limit of $\{s_n\}$. Thus, given any $\epsilon > 0$, there can only be finitely many terms as large as $m + \epsilon$.
I'm not sure if my understanding of the last sentence is totally accurate -
Is this is true because if there are infinite $n$ such that $m + \epsilon \leq s_n \leq S$, and since every bounded sequence has a convergent subsequence, then there would be some other subsequential limit greater than $m$, thus contradicting the initial conditions?
