A nonnegative real below every epsilon is zero

Lemma. Let $\ell\ge 0$ be a real number. If

then $\ell=0$.

Proof. If $\ell>0$, choose $\varepsilon:=\ell/2>0$. Then $\varepsilon<\ell$, contradicting the assumption that $\ell<\varepsilon$ for every $\varepsilon>0$. Hence $\ell=0$.

This lemma is commonly used to conclude equality from estimates that hold for all $\varepsilon>0$, e.g. in uniqueness of limits .