Science:Math Exam Resources/Courses/MATH220/April 2005/Question 05/Solution 1

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 $| \sum_{i = 1}^{n} a_{i} | \leq \sum_{i = 1}^{n} | a_{i} |$ holds for some (fixed) $n \in ℕ$ . Then we show that it also holds with $n$ replaced by $n + 1$ :

\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}

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 $n \in ℕ$ by induction.