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

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