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Math Help - Limit of two convergent sequences

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    Limit of two convergent sequences

    Let xn, yn be two convergent sequences. Show that limn-->∞xnyn exists.
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    Quote Originally Posted by ashamrock415 View Post
    Let xn, yn be two convergent sequences. Show that limn-->∞xnyn exists.
    Since the sequences converge, they have (finite) limits.

    Let \displaystyle \lim_{n \to \infty} x_n = x and \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 \lim_{n \to \infty} x_ny_n = xy

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

    I will give you a hint how to proceed.

    Note that \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?
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