> [!Lemma] Triangle inequality > For all $a,b \in\mathbb{R}$ we have that $|a+b|\leq |a|+|b|$with equality when $a,b \geq 0$ or $a,b \leq 0$. >*Proof*. For any $a,b \in\mathbb{R}$ we have $-|a|\leq a\leq |a|$ and the same for $b$, $-|b|\leq b\leq |b|$. It follows that $-|a|-|b|\leq a+b \leq |a|+|b| \iff |a+b|\leq|a| +|b|.$ >Now if $a,b \geq 0$ then $|a|=a$, $|b|=b$ and $|a+b| =a+b$. >If $a,b\leq 0$ then $|a|=-a$, $|b|=-b$ and $|a+b|=-(a+b)$. >If $a<0<b$, then $|a+b|=||b|-|a|| \neq |a|+|b|$ if $a,b \neq 0$. ### Applications - [[Reverse triangle inequality]]. - [[Algebra of Limits of Convergent Sequences]]. - [[Convergence]]. - [[Real Cauchy Sequence]]. - [[General Principle of Convergence]].