Calculus III - Multivariate limit problem

[Jonmundsson]

Homework Statement

Show that for all points (\bf{a,b}) from \mathbb{R}^n \times \mathbb{R}^n<br /> this applies:

\displaystyle \lim _{(\bf{x,y}) \to ({\bf a,b})} \bf{||x-y|| = ||a - b||}

Homework Equations

Not sure.

The Attempt at a Solution

I thought about defining a and b as centers of two open balls with x in the a ball and y in the b ball but honestly I'm stuck so any tips to help me get started would be appreciated.

Thanks.

edit: for clarification ||x-y|| means the distance between x and y.

 

[maCrobo]

I'm not sure it may help, but I would deal with it by simply applying the definition of distance, that should be the norm of those vectors. So I would define X and Y as vectors of n components and show clearly, maybe with a few passages how they go to a and b.

 

[JG89]

Does ||x-y|| represent the Euclidean norm, i.e. ||x-y|| = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2} where x = (x_1, x_2), y = (y_1, y_2)?

If so, the norm is clearly continuous. Give a quick proof of it then your proposition follows.

If it's not the Euclidean norm, you can still prove that a norm on any vector space is continuous, from which your proposition will again follow.

 

[Jonmundsson]

D'oh. It's Euclidean.

Thanks!