Relation between Supremum and limit superior

Question

If $\sup\limits_{n\ge 1} a_n<\infty$ then obtains that $\limsup\limits_{n\to\infty} a_n<\infty.$ Can you explain that?

Answer 1

We show something sharper. Denote $\;M=\sup_{n\ge1}a_n$ and suppose

$$B=\lim\sup_{n\to\infty}a_n>M\implies B=M+\delta\;,\;\;\delta>0$$

but then we can choose $\;\epsilon=\frac\delta2\;$ , and thus we obtain that there exists $\;n\in\Bbb N\;$ s.t.

$$|a_n-B|<\epsilon\implies a_n>B-\epsilon>B-\delta=M$$

which of course is absurd. In particular, it can't be $\;B=\infty\;$ .