$\definecolor{c1}{RGB}{255, 0, 0} \definecolor{c2}{RGB}{0, 0, 255} \definecolor{c3}{RGB}{255, 165, 0} \definecolor{c4}{RGB}{75, 0, 130} \definecolor{c5}{RGB}{220, 220, 0} \definecolor{c6}{RGB}{238, 130, 238} \definecolor{c7}{RGB}{0, 128, 0} \definecolor{c8}{RGB}{100,100,100} \definecolor{c9}{RGB}{45,177,93} \definecolor{default}{RGB}{10,20,20} \color{default}$ # introduction %%visits: 5 Cauchy [[convergence]] is helpful when you can’t exacly prove convergence directly. It helps to have it in your head so that you can easily use it to solve difficult problems. # diary # intuition Sequence converges if and only if it is a cauchy sequence that converges

rigour

Every convergent Sequence is a Cauchy sequence

let s_n be a convergent sequence with limit a then there exists an n₀ such that $\left| s_n - 1 \right| < \frac{1}{2}\epsilon$ for all n ≥ n₀

Now $\left| s_m - s_n \right| = \left| s_m - l + l - s_n \right| = \color{c1} \left| s_m - l + l - s_n \right| \le \left| s_m - l \right| + \left| l - s_n \right| = \left| s_m - l \right| + \left| s_n - l \right|$

The $\color{c1} \text{triangle inequality}$

So, if m, n ≥ n₀, we have $\left| s_m - l \right| < \frac{1}{2} \epsilon$,$\left| s_n - l \right| < \frac{1}{2} \epsilon$ and hence |s_m − s_n| < ϵ▫ # examples tags :math:sequences_and_series:real_analysis:

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