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Inequality with modulus

rayquaza17

I don't even know how to start. :frown:

(edited 9 years ago)

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Reply 1

Noble.

This is essentially an extension of the usual triangle inequality. This is one of the few times I advocate thinking about things geometrically ( :lol:

)

You can think of the LHS as a 'straight line' from x to y

The RHS is the journey: x to a, a to b, b to y.

The latter is only ever going to be as short as the straight line x to y if a and b lie on the straight line from x to y.

(Note, the algebraic proof to this is also straight forward, but thinking about it geometrically can help motivate how to start)

(edited 9 years ago)

Reply 2

Noble.

If that isn't helpful, the key hint is to use the triangle inequality.

Reply 3

username1162972

Original post by Noble.

If that isn't helpful, the key hint is to use the triangle inequality.

Is it possible with vectors?

Posted from TSR Mobile

Reply 4

TeeEm

Original post by rayquaza17

I don't even know how to start. :frown:

|x-y|=|x-a+a-b+b-y|=|(x-a) + (a-b) + (b-y)| <= .... by a famous result

Reply 5

rayquaza17

Original post by TeeEm

|x-y|=|x-a+a-b+b-y|=|(x-a) + (a-b) + (b-y)| <= .... by a famous result

What's the famous result?

Thanks, I had written down (x+a)+(b-a)-(y+b) [after seeing Noble's hint] and I was wondering why it didn't work.

Edit: I've now realised I had it written down like that because I had written the question down wrong. :facepalm:

Original post by Noble.

If that isn't helpful, the key hint is to use the triangle inequality.

Nope, it was helpful! Thank you!

I know it was such a simple question, but it's one of the first ones on a long list of analysis questions I need to do and I think because it's on that list that my mind is going blank. (Analysis and me don't really get on.)

(edited 9 years ago)

Reply 6

Noble.

Original post by physicsmaths

Is it possible with vectors?

Posted from TSR Mobile

Yes, you can do it with

\mathbb{R}^n

as well, from the same geometric principal.

Reply 7

Noble.

Original post by rayquaza17

What's the famous result?

Thanks, I had written down (x+a)+(b-a)-(y+b) [after seeing Noble's hint] and I was wondering why it didn't work.

Nope, it was helpful! Thank you!

I know it was such a simple question, but it's one of the first ones on a long list of analysis questions I need to do and I think because it's on that list that my mind is going blank. (Analysis and me don't really get on.)