$\inf f(A) \leq f( \inf A)$ if $f$ is continuous

Question

Prove that $\inf f(A) \leq f( \inf A)$ if $f: [-\infty, + \infty] \to \mathbb{R}$ is continuous and $A \neq \emptyset$ is a subset of $\mathbb{R}$.

Attempt;

Put $a:= \inf A$. Choose a sequence in $A$ such that $a_n \to a$. Then

$$ \inf f(A)\leq\lim_{n \to \infty} \underbrace{f(a_n)}_{\geq \inf f(A)} = f(a) = f( \inf A)$$

and we can conclude.

Is this correct?

Answer 1

Since the domain of $f$ is $[-\infty,\infty]$, something is worth to mention.

For nonempty subset $A$ of $\mathbb{R}$, $\inf A\ne\emptyset$. But it could be the case that $\inf A=-\infty$. So a sequence $(x_{n})\subseteq A$ is such that $x_{n}\rightarrow-\infty$, $f$ being continuous at $-\infty$, we still have $f(x_{n})\rightarrow f(-\infty)$.

Answer 2

You have $\inf_x f(x) \le f(y)$ for all $y$ by definition. Ley $y = \inf A$ to finish.