- The problem statement, all variables and given/known data
I understand the first bound but not the second.
I am fine with the rest of the derivation that follows after these bounds, 2. Relevant equations
I have this as the triangle inequality with a '+' sign enabling me to bound from above:
$|x+y| \leq |x|+|y| $ (1) $|x-y| \geq |x|-|y| $ (2)
and this as the triangle inequality with a '-' sign enabling me to bound from below:
- The attempt at a solution
So for the first bound we have:
$|z-w| \geq |z| - |w| $
since we have a strict less than inequality for |z| and a strict greater than equality for |w| , both of these are consistent and we indeed loose the equality option in the triangle inequality to get $|z-w| > -R $
I am stuck on the second bound however. 1) I ionly have a upper bound for a subtraction and not a lower via the triangle inequalities. can i get a upper bound from (1) and (2)?
i.e. are you allowewd to do $|z+(-2w)| \leq |z|+|-2w| $?
(even if I am, unlike the lower bound, where it turns out the bound we have on $z$ and $w$ are consistent with the triangle inequality, (enabling us to loose the equality and get strictness) there is contrast between the inequalities in this case. ( both items on the right hand side would need lower bounds (or one upper and one equality) but z has a upper bound).