Prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$ I'm trying to prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$
I've first written down $-5\lt a-b \lt5$ and have tried to add different things from all sides of the inequality. Like adding $b+5$ to get $b\lt a+5 \lt 10+b$ but am just not seeing where that gets me.

We have that $$|a-b| \leq 5$$ 
It stands that $$||a|-|b|| \leq |a-b| \\ \Rightarrow -|a-b| \leq |a|-|b| \leq |a-b|$$ 
From the inequalities $-|a-b| \leq |a|-|b|$ and $|a-b| \leq 5$ we get $$|b|-|a| \leq |a-b| \leq 5 \Rightarrow |b|-|a| \leq 5 \Rightarrow |b| \leq |a|+5$$ 
