Clarification on the proof that all bounded monotone sequences converge.

Question

In the second edition of Elementary Analysis by Ross in the proof for the theorem that states all bounded monotone sequences converge they have the following statement in their proof.

$U - \epsilon < s_n \leq U \implies |s_n - U| < \epsilon$

This is more of a clarification on the inequality. Wouldn't we need the inequality to be $U - \epsilon < sn < U + \epsilon$ to make the conclusion that $|s_n - U| < \epsilon$?

Answer 1

If $s_n\leq U$, then $s_n < U+\epsilon$, provided $\epsilon > 0$, so the right side of the original inequality can be replaced by $s_n<U+\epsilon$.