If $|a+b|≤1,$ then $ |a|≤|b|+1.$

Question

How can I prove

If $|a+b|≤1,$ then $|a|≤|b|+1$

in real analysis ?

I try to use Triangle inequality

Answer 1

$|a|=|a+b-b|\le |a+b|+|-b|=|a+b|+|b|\le 1+|b|$

Answer 2

Hint: $$ a=a+b+(-b). $$ You should be able to go from here.

Answer 3

$$|a|-|b|\leq |a+b|\leq1\implies |a|\leq |b|+1$$

You can also remark that $|b|\leq |a|+1$.

Answer 4

We have $|a+b|-|b|\leq |a|$.

Now set $a=x+y$ and $b=-y$

This yields $|x|-|y|\leq |x+y|$

Answer 5

See that it is true: |a|-|b|<|a+b|≤1.