Calculus III - Multivariate limit problem

  • #1

Homework Statement


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

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

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.
 

Answers and Replies

  • #2
51
1
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.
 
  • #3
726
1
Does [itex]||x-y|| [/itex] represent the Euclidean norm, i.e. [itex] ||x-y|| = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2} [/itex] where [itex] x = (x_1, x_2), y = (y_1, y_2) [/itex]?

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.
 
  • #4
D'oh. It's Euclidean.

Thanks!
 

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