Science:Math Exam Resources/Courses/MATH220/April 2005/Question 05

MATH220 April 2005

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Question 05
Let $a_{1},a_{2},a_{3},\ldots$ be real numbers. Prove, using induction, that for all $n\in \mathbb {N}$, $\left|\sum _{k=1}^{n}a_{k}\right|\leq \sum _{k=1}^{n}|a_{k}|$ You may assume the Triangle Inequality in the form $|x+y|\leq |x|+|y|$ for all real numbers x and y.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Use that $\sum _{i=1}^{n}a_{i}=\sum _{i=1}^{n-1}a_{i}+a_{n}$

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We use induction. In the base case when $n=1$ we have $|a_{1}|\leq |a_{1}|$ so the statement is true. Induction step: we assume that $|\sum _{i-1}^{n}a_{i}|\leq \sum _{i=1}^{n}|a_{i}|$ holds for some (fixed) value $n\in N$. Then it also holds for $n+1$: ${\begin{aligned}|\sum _{i-1}^{n+1}a_{i}|&=|\sum _{i-1}^{n}a_{i}+a_{n+1}|\\&\leq |\sum _{i-1}^{n}a_{i}|+|a_{n+1}|\\&\leq \sum _{i=1}^{n}|a_{i}|+|a_{n+1}|\\&=\sum _{i=1}^{n+1}|a_{i}|\end{aligned}}$ Where the first inequality is true by the triangle inequality and the second by induction hypothesis. This finishes the induction step and so the result is true for all $n\in N$ by induction.

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