Infimum

Let $S$ be a set of real numbers. If there exists a real number $b$ such that $x\ge b$ for every $x\in S$, then $b$ is called a lower bound for $S$ and we say that $S$ is bounded below by $b$.

• We say “a” lower bound because any number less than $b$ will also be a lower bound.

• If $b$ is also an element of $S$ then it is called the minimum element of $S$.

• A set with no lower bound is called unbounded below.

The infimum of the set $S$ is the greatest of all the lowerbounds of the set. It is denoted as $inf(S)$