I appreciate any help you can give me in solving this problem.
Let $x\in\mathbb{R}$ and assume that for all $\epsilon > 0$, $|x|< \epsilon$. Prove that $x = 0$. Use a proof by contradiction.
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0Hint: What about $\epsilon=|x|$? – 2017-02-16
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1The body of the Question is intended to allow a self-contained problem statement. Relying on the title alone to bear this burden invites omission of setup and other context, leading to confusion for Readers. – 2017-02-16
1 Answers
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Suppose $x\neq0$ and take $\epsilon=\frac{\vert x\vert}{2}$. Then you see that $\vert x\vert\ge\epsilon$ (and even $>$).
Hence $x=0$.