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 the fact that $\sum _{i=1}^{n}a_{i}=\sum _{i=1}^{n-1}a_{i}+a_{n}$.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. In the base case when $n=1$, we have $|a_{1}|\leq |a_{1}|$, so the statement is true. For the inductive step, we assume that $\left|\sum _{i=1}^{n}a_{i}\right|\leq \sum _{i=1}^{n}|a_{i}|$ holds for some (fixed) $n\in \mathbb {N}$. Then we show that it also holds with $n$ replaced by $n+1$: ${\begin{aligned}\left|\sum _{i=1}^{n+1}a_{i}\right|&=\left|\sum _{i=1}^{n}a_{i}+a_{n+1}\right|\\&\leq \left|\sum _{i=1}^{n}a_{i}\right|+|a_{n+1}|\\&\leq \sum _{i=1}^{n}|a_{i}|+|a_{n+1}|\\&=\sum _{i=1}^{n+1}|a_{i}|.\end{aligned}}$ The first inequality is true by the triangle inequality and the second by the induction hypothesis. This finishes the inductive step and so the result is true for all $n\in \mathbb {N}$ by induction.

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