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