Let $(a_{n})_{n \geq1}$ be a sequence of numbers such that $a_n\leq M$ for all $n \geq 1$ . Prove that
$$ \lim_{n\to\infty} \inf \{a_n,a_{n+1},...\} = \sup_{n \geq1} \inf\{a_n,a_{n+1},...\} $$
This is a homework problem.
But I'm really confused where to start. I'm an engineering student taking analysis course and have no previous background of rigorous proof. It'll be help full if someone can tell me how to approach the problem.
Thank you.
Consider the sequence of numbers $$ I_1=\inf\{a_1,a_2,\ldots, a_n,\ldots\}\leq a_1\leq M, \\ I_2=\inf\{a_2,a_3,\ldots, a_n,\ldots\}\leq a_2\leq M, \\ \vdots \\ I_n=\inf\{a_n,a_{n+1},a_{n+3}\ldots\}\leq a_n\leq M. $$ form non-empty sets $\{I_1,\ldots, I_n\}$, for each $n\in\mathbb{N}$, bounded by constant $M$. By definition, we have the supreme of the set is such that $$ \sup\{I_1,\ldots,I_n\}\geq I_n. $$ On the other hand, as $ I_1\leq I_{2}\leq \ldots \leq I_{n}\leq \ldots \leq M $ we have $$ \sup_{n\geq 1}\inf\{a_n,a_{n+1},a_{n+2},\ldots \}=\sup_{n\geq 1}\{I_1,\ldots, I_n\}=I_n\leq M. $$ So what can you say about $\lim_{n\to \infty}S_n$,for $ S_n=\sup\{I_n, I_{n+1},I_{n+2},\ldots \}, $ and $\lim_{n\to\infty}I_n$ ?