Exercise 1864Logic, sets and proofs › Quantifiers and negationL1Added 3 Nov 2026

A statement with a specific structure

#epsilon#forall#exists

Statement

Prove: $\forall\varepsilon>0,\ \exists\delta>0,\ \forall x\in(0,\delta),\ \sqrt{x}<\varepsilon$ .

Hint

Take $\delta=\varepsilon^2$ .

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Choice. For $\varepsilon>0$ , choose $\delta=\varepsilon^2>0$ .

Verification. For $x\in(0,\delta)=(0,\varepsilon^2)$ , $\sqrt x<\varepsilon$ . ✓

Conclusion. The statement holds; this proves $\lim_{x\to0^+}\sqrt x=0$ .