A question about infimum: $k\leq a_i\implies k\leq\inf_i a_i$?

Question

Let $k$ be a constant real number, let $A\subseteq \mathbb{R}$ be a set with countable cardinality.

if $k\leqslant a$ for all $a\in A$, is it true that $k\leqslant \inf A$?

Must we have $k<a$ for all $a$?

Answer 1

Suppose $A$ is a non empty set such that $k \le a$ for all $A$. Then $k \le \inf A$.

If $\inf A < k$ then there is some $a \in A$ such that $a < {1 \over 2} (k+\inf A)$, which gives $a < k$, a contradiction.

Take $k=0$, $A=\{0,1,2,...\}$. Then $k = \inf A$.