How can i prove that $a_{n} < L + \epsilon$?

Question

given $a_{n}$ bounded series, and L= $lim sup(a_{n})$.

How can i prove that: for any $\epsilon > 0$ exist $n_{0}∈N $ , so that for any $n∈N$ :
if $n>n_{0}$ so: $a_{n} < L + \epsilon$.

Answer 1

Suppose some $\epsilon>0$ exists for which no $n_0\in\mathbb N$ exists with $n>n_0\implies a_n<L+\epsilon$.

Then a subsequence $(a_{n_k})_k$ can be shown to exists with $a_{n_k}\geq L+\epsilon$ for every $k$.

This implies that $\limsup a_n\geq L+\epsilon$ and a contradiction is found.