Show that $|a|>|b|/2$ knowing that $|a−b|<|b|/2$.

Question

I'm working on a proof and I need to show that $|a|>|b|/2$ knowing that $|a-b|<|b|/2$. I would like to do it without enumerating the different cases.

Thanks for you ideas.

Answer 1

Using the triangle inequality in the 2nd inequality below: $$ \frac{|b|}{2}>|a-b|=|b-a|\geq|b|-|a|\implies |a|>|b|-\frac{|b|}{2}=\frac{|b|}{2}. $$

Edit: a few more details: the triangle inequality states that: $$ |x+y|\leq|x|+|y|. $$ So applying it with $x=b-a$ and $y=a$ yields $$ |b|=|b-a+a|\leq |b-a|+|a|\implies|b-a|\geq|b|-|a|. $$