Let xn, yn be two convergent sequences. Show that limn-->∞xnyn exists.

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- Oct 5th 2010, 06:18 PMashamrock415Limit of two convergent sequences
Let xn, yn be two convergent sequences. Show that limn-->∞xnyn exists.

- Oct 5th 2010, 06:43 PMJhevon
Since the sequences converge, they have (finite) limits.

Let $\displaystyle \displaystyle \lim_{n \to \infty} x_n = x$ and $\displaystyle \displaystyle \lim_{n \to \infty} y_n = y$

We will not only show that the limit of the product exists, but we will say what it is! We claim that

$\displaystyle \displaystyle \lim_{n \to \infty} x_ny_n = xy$

We need to show that for every $\displaystyle \displaystyle \epsilon > 0$, there exists an $\displaystyle \displaystyle N \in \mathbb N$ such that $\displaystyle \displaystyle n > N$ implies that $\displaystyle \displaystyle |x_ny_n - xy| < \epsilon$

I will give you a hint how to proceed.

Note that $\displaystyle \displaystyle |x_ny_n - xy| = |x_ny_n - x_ny + x_ny - xy| \le |x_n||y_n - y| + |y||x_n - x|$

Now what can you say?