$A$ is a non-empty subset that is lower bounded. Let $B=A\cap(-\infty, m+1)$ and $m=\inf A$. Prove $B$ is nonempty and lower bounded.

Question

$A$ is a non-empty subset that is lower bounded. Let $B=A \cap(-\infty, m+1)$ and $m=\inf A$.
Prove $B$ is nonempty and lower bounded.

I'm having difficulties finding a starting point for this question and would like a hint. Thanks.

Answer 1

Hint for first part:

Suppose $B$ is empty. Then there does not exist $a\in A$ such that $a<m+1$, i.e. for all $a\in A$ ...

Hint for second part:

You know that $A\subseteq [m,\infty)$. Thus $B=A\cap (-\infty,m+1]\subseteq $...