Proof By Induction for arbitrary integers

Question

Assume that $|x + y| \leq |x| + |y|$ for all $x,y \in {Z}.$ Use this assumption and induction to prove that

$$|a_1 + a_2 + ... + a_n| \leq |a_1| + |a_2| + ... + |a_n|$$

for all integers $n \geq 2$ and arbitrary integers $a_1, a_2,..., a_n.$

Answer 1

If $n=2$ then $|a_{1}+a_{2}| \leq |a_{1}| + |a_{2}|$ for all integers $a_{1},a_{2}$ by assumption.

Suppose $n \geq 2$ such that $|a_{1}+\cdots + a_{n}| \leq |a_{1}| + \cdots + |a_{n}|$ for all integers $a_{1},\dots, a_{n}$. Since $|a_{1}+\cdots + a_{n+1}| \leq |a_{1} + \cdots + a_{n}| + |a_{n+1}|$ for all integers $a_{1},\dots, a_{n+1}$ by the case where $n=2$, and since by induction hypothesis we have $|a_{1} + \cdots + a_{n}| \leq |a_{1}| + \cdots + |a_{n}|$ for all integers $a_{1},\dots, a_{n}$, it follows that $$ |a_{1} + \cdots +a_{n+1}| \leq |a_{1}| + \cdots + |a_{n+1}|. $$ Then by mathematical induction we have are done.