Triangle Inequality: use to prove convergence

[binbagsss]

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 .

 

[andrewkirk]

i.e. are you allowewd to do $|z+(-2w)| \leq |z|+|-2w|$?

Yes that is a standard application of the triangle inequality.

(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).

Note that the lower bounds are to be expressed as multiples of $|\omega|$. It is trivial to get such a lower bound for the second term on the RHS. The two inequalities given in the problem enable us to get a lower bound in terms of $|\omega|$ for the first term.

 

[binbagsss]

Yes that is a standard application of the triangle inequality.
Yes that is a standard application of the triangle inequality.

Ok thanks so from similar manipulations with (2) I see that $|x+y|$ and $|x-y|$ have the same upper and lower bounds from the triangle inequality which makes sense when I think about what the mod function does and possible combinations etc.

The two inequalities given in the problem enable us to get a lower bound in terms of $|\omega|$ for the first term.

That is fine.

Note that the lower bounds are to be expressed as multiples of $|\omega|$. It is trivial to get such a lower bound for the second term on the RHS.

I still don't understand this. Why is it trivial?
As I said in my OP 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.

 

[andrewkirk]

I still don't understand this. Why is it trivial?

We are looking for upper bounds, not lower bounds.
We want to find an upper bound for $|z|+|-2w|$ that is no greater than $\frac52|w|$.

The second term is $|-2w|$ and we want to find an upper bound for it that is a multiple of $|w|$. Given that an equality is also a upper bound, can you express $|-2w|$ as a multiple of $|w|$ and thereby have an upper bound for $|-2w|$?

For the first term $|z|$, the two inequalities you've been given are $|z|<R$ and $|w|>2R$. How can you use those to get an upper bound for $|z|$ that is a multiple of $|w|$?

 

[binbagsss]

We are looking for upper bounds, not lower bounds.
We want to find an upper bound for $|z|+|-2w|$ that is no greater than $\frac52|w|$.

The second term is $|-2w|$ and we want to find an upper bound for it that is a multiple of $|w|$. Given that an equality is also a upper bound, can you express $|-2w|$ as a multiple of $|w|$ and thereby have an upper bound for $|-2w|$?

For the first term $|z|$, the two inequalities you've been given are $|z|<R$ and $|w|>2R$. How can you use those to get an upper bound for $|z|$ that is a multiple of $|w|$?

No but the only information is $|w|>2R$, $|z|<R$ , I can see clearly that the aim is to express in multiplies of $w$ and I'm pretty sure I said I UNDERSTAND the first inequality so unsure why everyone is replying to that, but anyway, given this all we know is $|w|>2R$ , it could be infinity, that's all we know? how do we bound that?

 

[andrewkirk]

No but the only information is $|w|>2R$, $|z|<R$

Put those two together to get an inequality relation between |z| and |w|. Then use that to get an upper bound on |z|+|-2w| (first converting the second term of that into a multiple of |w|).

 

[binbagsss]

Put those two together to get an inequality relation between |z| and |w|. Then use that to get an upper bound on |z|+|-2w| (first converting the second term of that into a multiple of |w|).

I don't believe I'd have any issue with going from a bound between |z| and |w| to one of |z|+|-2w| . My issue in my previous still stands whether I am upper bounding - I assume this is what you meant by an inequality - |z|+|-2w| or |z|+|w| - as far as I can see z can be as large as it likes and so I can't see how we can upper bound anything involving an addition of z. Ta.

 

[binbagsss]

I don't believe I'd have any issue with going from a bound between |z| and |w| to one of |z|+|-2w| . My issue in my previous still stands whether I am upper bounding - I assume this is what you meant by an inequality - |z|+|-2w| or |z|+|w| - as far as I can see z can be as large as it likes and so I can't see how we can upper bound anything involving an addition of z. Ta.

?

 

[LCKurtz]

$$|z| < R < 2R < |\omega| \text{ ?}$$

 

[binbagsss]

$$|z| < R < 2R < |\omega| \text{ ?}$$

Apologies typo, \omega can be as large as it wants