Math Help - Convergent sequences

1. Convergent sequences

let

2. Originally Posted by Godisgood
Let X_n and y_n be two convergent sequence in the metric space (X,d) with X_n converging to x and y_n converging to y. show that d(x_n, y_n) converges to d(x,y)

Thanks

$d(x_n,y_n) \le d(x_n,x)+d(x,y_n)\le d(x_n,x)+d(x,y)+d(y,y_n)$

This implies

$d(x_n,y_n)-d(x,y) \le d(x_n,x)+d(y,y_n)$

This should get you started. This is half of the inequality.

3. That is true only if d(*,*) is continous in [x,y]...

Kind regards

$\chi$ $\sigma$

4. Do you know that $\left| {d(x,y) - d(x_n ,y_n )} \right| \leqslant \left| {d(x,y) - d(x_n ,y)} \right| + \left| {d(x_n ,y) - d(x_n ,y_n )} \right|~~(\#1)$

There is a well-known theorem for metric spaces:
for any points $a,~b~\&, t \in S$ it is the case that $\left| {d(a,t) - d(b,t)} \right| \leqslant d(a,b)$.

Apply that to both terms on the right of $~(\#1).$