- #1

binbagsss

- 1,260

- 11

## Homework Statement

Attached

I understand the first bound but not the second.

I am fine with the rest of the derivation that follows after these bounds,

## Homework 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).

Many thanks .