Let xn, yn be two convergent sequences. Show that limn-->∞xnyn exists.
Since the sequences converge, they have (finite) limits.
Let and
We will not only show that the limit of the product exists, but we will say what it is! We claim that
We need to show that for every , there exists an such that implies that
I will give you a hint how to proceed.
Note that
Now what can you say?