Question: (a) Give the definition that a sequence \(\left\{a_{n}\right\}\) is a Cauchy sequence.

(b) Let \(f\) be a differentiable function on \((0,1)\) such that \(\left|f^{\prime}(x)\right| \leq 1 .\) Use the mean value theorem to show that, for every natural number m, n,

(c) Let \(f\) be the same as in (b). Let \(a_{n}=f\left(\frac{1}{n}\right)\). Prove that \(\left\{a_{n}\right\}\) is a Cauchy sequence (Hint: use (b))

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