1
$\begingroup$

I'm given this

Prove that if $\lim\limits_{n \to \infty} x_n = x $ & $\lim\limits_{n \to \infty} y_n = y$ then $\lim\limits_{n \to \infty}d(x_n,y_n)$ = d(x,y)

Here's what I know, d(x,y) = |x-y|

So far I've tried writing out the epsilon definitions of limits for $x_n$ and $y_n$ but I'm just not having any luck with this proof. Any pointers?

$\endgroup$
4
$\begingroup$

Use the inequalities that $|d(x_{n},y_{n})-d(x,y)|\leq|d(x_{n},y_{n})-d(x_{n},y)|+|d(x_{n},y)-d(x,y)|\leq d(y_{n},y)+d(x_{n},x)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.