What does $\inf A=\infty$ mean?

Question

I'm confused with the meaning of $\inf A=\infty$ for $A\subset \mathbb R.$

I know that $m=\inf A$ $\iff$ (i) $\forall a\in A, m\leqq a \ $ (ii) $\forall \epsilon>0, \exists a\in A$ s.t. $m+\epsilon >a$

If $\inf A=\infty,$ then $m=\infty$ and (i) is $\forall a\in A , \infty \leqq a$.

This means $\forall a\in A, a=\infty$ and thus $A=${$\infty$}, but this is unnutural to me. (because $A=\{\infty\}\not\subset \mathbb R$)

So, I wonder which is the meaning (or definition) of $\inf A=\infty.$

Is it $\forall M>0, \inf A>M$ ?

Answer 1

For subsets of $\mathbb{R}$ the only way for the inf to be $\infty$ is for the set to be empty, in which case any real number is a lower bound and so the supremum of all the lower bounds is $\infty$. The same applies for the sup being $-\infty$. For a nonempty set the inf is either finite (if there are any real lower bounds) or $-\infty$ (otherwise).