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A nonnegative real below every epsilon is zero

A nonnegative real number smaller than every positive real number must be zero.

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

ℓ<εfor every ε>0,\ell<\varepsilon \quad\text{for every }\varepsilon>0,

then ℓ=0\ell=0.

Proof. If ℓ>0\ell>0, choose ε=ℓ/2\varepsilon=\ell/2. Then ε<ℓ\varepsilon<\ell, contradicting the hypothesis. Hence ℓ=0\ell=0.

Remarks

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