Proper Divergence and Limits

**jkru** · November 5th 2008, 06:29 AM

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

**Laurent** · November 5th 2008, 09:26 AM

Originally Posted by jkru

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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