11. (3 points) Use contradiction to prove the following statement $\forall x \in \mathbb{R}$, If $|x| < \varepsilon$ for any $\varepsilon > 0$, then $x = 0$.
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Since x ≠ 0, we can consider two cases: 1) x > 0 2) x < 0 Case 1: x > 0 In this case, let's choose e = x/2. Since x > 0, e > 0. But |x| < e implies |x| < x/2. Multiplying both sides by 2, we get 2|x| < x. This contradicts the assumption that x > 0. Show more…
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