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Math Help - Proper Divergence and Limits

  1. #1
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    Proper Divergence and Limits

    Let Xsubn by properly divergent and let Ysubn be such that lim(Xsubn*Ysubn) belongs to the reals. Show that Ysubn converges to 0. I'm not really sure how to go about proving this. I know that Ysubn must converge, I'm not sure how to show that it converges to 0. Any help is much appreciated. Thanks
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  2. #2
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    Quote Originally Posted by jkru View Post
    Let (x_n)_n by properly divergent and let (y_n)_n be such that x_n y_n \xrightarrow[n\to\infty]{}\ell\in\mathbb{R}. Show that (y_n)_n converges to 0. I'm not really sure how to go about proving this. I know that (y_n)_n must converge, I'm not sure how to show that it converges to 0. Any help is much appreciated. Thanks
    Procede by contradiction: Suppose (y_n)_n does not converge to 0. Then there exists a real number \varepsilon>0 and a subsequence (y_{\varphi(n)})_n such that, for every n, |y_{\varphi(n)}|>\varepsilon.
    However, this implies |x_{\varphi(n)}y_{\varphi(n)}|\geq \varepsilon|x_{\varphi(n)}|\xrightarrow[n\to\infty]{}+\infty, in contradiction with the fact that (x_ny_n)_n converges in \mathbb{R}.
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