Epsilon Principle

Last updated Nov 1, 2022

Let $a,b \in \mathbb{R}$. Suppose for all $\epsilon > 0$, we have that $a \geq b - \epsilon$. Then $a \geq b$.

Suppose not. Then $a < b$ and there exists some $\delta > 0$ s.t. $a = b - \delta$. But then $a < b - \frac{\delta}{2}$ which contradicts our assumption that $a \geq b - \epsilon$ for all $\epsilon > 0$. $\blacksquare$

• Real Numbers