Infimum of a sequence bounded from below

Question

Suppose a sequence $a_n\geq b$ for all $n$, is it true that $\inf a_n\geq b$? My suggestion is yes. Suppose $\inf a_n$ exist and is equal $c$. Then according to the definition of $\inf$, there exist a $\epsilon>0$ s.t. $c+\epsilon$ contains infinitely many member of $a_n$. Suppose $c=\inf a_n< b$, then there exist a $\epsilon>0$ (by definition of open set) s.t $c+\epsilon<b$. Hence, there must be infintely member of $a_n$ between $[c,c+\epsilon]$ which definitely is less than b and this contradicts to the assumption $a_n\geq b$ for every n.

Answer 1

It is simpler than this. By your hypothesis, $b$ is a lower bound for $\{a_n\}$. By definition, $\inf A$ is at least as great as each lower bound of $A$, so $\inf\{a_n\}\geq b$.