Mathematical Formulation - No number less than Supremum M can be an upper bound

Question

I was reading up on supremum and infimum and came across a line whose mathematical formulation has confused me. It says, "No number less than the supremum, M of a set S can be an upper bound of S." This line is very clear. Since M is itself the smallest member of the set of all upper bounds of S, this line is only obvious. But I am confused with the mathematical formulation that follows -

For any positive number , epsilon, however small, there exists a y belonging to S such that y > M - epsilon

Can someone help me with this?

Attaching an image of it as well.

Answer 1

If there were no $y$ in $S$ such that $y>M-\varepsilon$, y.e. if $y\le M-\varepsilon$ for all $y\in S$, $M-\varepsilon$ would be a smaller upper bound.