$|a-b|

Question

For $a,b\in\Bbb R$ prove $|a-b|<E$ iff $a-E<b<a+E$.

I am unsure how to start this proof. I think it might have something to do with the reverse triangle inequality but I am stuck.

Answer 1

Think of it this way: $|x|$ gives us the distance of a real number $x$ from $0,$ so when we say that $|x|<E$ for some (necessarily positive) $E,$ we are saying that $x$ is less than $E$ away from $0.$ Visualizing this on the number line, it means precisely that $x$ lies between $-E$ and $E.$ Can you translate this into an inequality (without the absolute value bars) and apply it to your problem?