Why $\inf \{f^2(x) : x \in A\} = [\inf \{f(x) : x \in A\}^2]$

Question

I am trying to prove the integrability of $fg$ and I found that in most proof, they claim that $$\inf \{f^2(x) : x \in A\} = \inf \{f(x) : x \in A\}^2$$ without proof and I am wondering how this can be proved. Thank you

Answer 1

This is false in general: suppose that $A=[-1,1]$ and $f(x)=x$. Then $$ \inf\{x^2:x\in [-1,1]\}=0 $$ while $$ (\inf\{x:x\in[-1,1]\})^2=(-1)^2=1$$

However, the claim is true if $f$ is a non-negative function.