If $\inf A < 1 $ ...?

Question

I just want to know if, in general, $\inf A < 1 $ implies that at least an element of $A$ is less then $1$.

$$ \inf A < 1 \rightarrow \exists\alpha \in A : \alpha < 1 ? $$

$A \subset \mathbb{R}$ and is obviously bounded from below.

Answer 1

Indeed, it does. If it were not true, then either $A=\emptyset$--in which case $\inf A=+\infty$ by convention--or $A$ is non-empty and every element of $A$ is at least $1,$ meaning that $1$ is a lower bound of $A,$ so that $1\le\inf A$ by definition of infimum.

Addendum: If we adopt the convention that $\inf A=-\infty$ when $A\subseteq\Bbb R$ is without lower bound, along with the convention $\inf\emptyset=+\infty$ mentioned above, then for any sets $A\subseteq\Bbb R,$ the following statements are equivalent:

• $\inf A<1$
• $\exists\alpha\in A:\alpha<1$

There is actually no need to make the assumption that $A$ is bounded below (though, as you say, the first statement certainly implies that this is true). More generally, we can replace $1$ with any real $\beta,$ and we can replace $\inf$ with $\sup$ if we reverse the inequalities--using the conventions $\sup\emptyset=-\infty$ and $\sup A=+\infty$ when $A\subseteq\Bbb R$ is unbounded above. For more on the intuition behind/connections between the four conventions I mentioned herein, you can see the answers here or here.

Answer 2

You have a subset $A$ of the real numbers. Assume that all the elements of $A$ satisfy that they are greater than or equal to $1$, i.e. that $A \subseteq [1,\infty)$.

Then what can you say about the infimum?