I know that there are several questions already about the following equality, however there is no, that helps me with my question.. I want to show the following:
Let $(s_n)$ be a sequence of real numbers, not necessarily bounded. $$\lim_{n \to \infty} \inf s_n = \lim_{n \to \infty} \inf_{k \geq n} s_k, $$ where $\lim_{n \to \infty} \inf s_n=inf(A)$, with $A$ the set of accumulation points of $s_n$.
My attempt: Suppose that $\lim_{n \to \infty} \inf s_n=:s < \infty$. We define the help sequence $b_n:= \inf_{k \geq n} s_k$. It follows that $b_n$ is increasing. It is not hard to see that we have $s \leq l:= \lim_{n \to \infty} b_n$. But how can I conclude the other inequality and use that my sequence $b_n$ is increasing. I can not just conclude that since $s$ is finite, my sequence $b_n$ has to be bounded. Else I could follow that with Bolzano Weierstrass there is a convergent subsequence.. Thanks for any help!