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anonymous

  • 5 years ago

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.

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  1. anonymous
    • 5 years ago
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    Does this seem sound?

  2. anonymous
    • 5 years ago
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    o..m..g..

  3. LifeIsADangerousGame
    • 5 years ago
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    I agree with Hershey_Kisses...

  4. anonymous
    • 5 years ago
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    holy cow! that looks like.....holy cow that unbearable!

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