# subsequences in R^n

• Oct 22nd 2008, 07:55 PM
squarerootof2
subsequences in R^n
i'm having difficulty proving the following:
let p_i(i in naturals) be a sequence of pts in R^n and p_0 be a point in R^n. let p_i=(p_i,1 , ... , p_i,n) for i in naturals and p_0 = (p_0,1 , ... , p_0,n). show that lim(i goes to infinite) p_i = p_0 if and only if lim(i goes to infinite) p_i,k = p_0,k for every k=1,...,n.
• Oct 22nd 2008, 08:09 PM
ThePerfectHacker
Quote:

Originally Posted by squarerootof2
i'm having difficulty proving the following:
let p_i(i in naturals) be a sequence of pts in R^n and p_0 be a point in R^n. let p_i=(p_i,1 , ... , p_i,n) for i in naturals and p_0 = (p_0,1 , ... , p_0,n). show that lim(i goes to infinite) p_i = p_0 if and only if lim(i goes to infinite) p_i,k = p_0,k for every k=1,...,n.

I will do it for $n=2$ but it generalizes.

Let $\bold{x}_n = (a_n,b_n)$ and $\bold{x}_0 = (a_0,b_0)$.

If $\bold{x}_n \to \bold{x}_0$ then $|(a_n,b_n) - (a_0,b_0)|$ can be made arbitrary small.
Thus, $|a_n-a_0|,|b_n-b_0|\leq \sqrt{(a_n-a_0)^2+(b_n-b_0)^2} < \epsilon$.

And if $|a_n - a_0|,|b_n-b_0| < \epsilon$ then $\sqrt{(a_n-a_0)^2+(b_n-b_0)^2} \leq \epsilon \sqrt{2}$.
Thus, $\bold{x}_n \to \bold{x}_0$.

This is Mine 11:):):)th Post!!!