## infimum of set of real numbers bounded below

Let $$M$$ be a sequence of real numbers that is bounded below. Show that it has a (real) infimum. (You may use the existence of a supremum for a set of numbers that is bounded above.)
The set $$-M := \{-x \colon x \in M\}$$ is bounded above, so it has a supremum $$s$$. Then $$-s$$ is an infimum of the set $$M$$.