• anonymous
Use the Triangle Inequality and Mathematical Induction to show that$\left|\sum_{k=1}^{n}a_k\right|\leq\sum_{k=1}^{n}|a_k|.$Well, from the Triangle Inequality, we know that$\left|\sum_{k=1}^{n}a_k\right|=|a_1+a_2+...+a_n|\leq|a_1|+|a_2|+...+|a_n|=\sum_{k=1}^{n}|a_k|,$and by Mathematical Induction, we see that$\left|\sum_{i=1}^{1}a_i\right|=|a_1|=\sum_{i=1}^{1}|a_i|,$assume it's true for $$k$$, and see that$\left|\sum_{i=1}^{k+1}a_i\right|=|a_1+a_2+...+a_{k+1}|\leq|a_1|+|a_2|+...+|a_{k+1}|=\sum_{i=1}^{k+1}|a_i|,$as required.
Mathematics

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