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Suppose $(x_n)$ and $y_n$ are two convergent sequences such that infinitely many of the terms satisfy $x_n \leq y_n$. Prove that $\displaystyle\lim_{n \rightarrow \infty}x_n \leq \displaystyle\lim_{n \rightarrow \infty}y_n.$

My attempt: Since $x_n \leq y_n$ holds for infinitely many $n$, by choosing subsequence, we can assume that $x_n \leq y_n$ holds for all $n$. We can also assume that $(x_n)$ and $(y_n)$ are convergent. Therefore, $\displaystyle\lim_{n \rightarrow \infty} x_n \leq \displaystyle\lim_{n \rightarrow \infty} y_n.$

Is my attempt correct?

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  • $\begingroup$ Using argument by contradiction. $\endgroup$ – xpaul Apr 21 '17 at 12:57
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    $\begingroup$ Your attempt looks pretty good indeed, yet instead of "We can also assume...", you could more accurately write "We know that ..." . Also, a sequence converges iff any subsequence converges to same limit as the sequence $\endgroup$ – DonAntonio Apr 21 '17 at 13:02
  • $\begingroup$ Your attempt is quasi-circular, you just traded "infinitely many" for "all", but the main claim remains unchanged. $\endgroup$ – Yves Daoust Apr 21 '17 at 13:28
  • $\begingroup$ You are wrong; one can be less than the other for infinitely many terms but still not be for all terms. $\endgroup$ – Jacob Wakem May 10 '17 at 20:23
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Hint, by contraposition:

  • suppose that $\displaystyle\lim_{n \rightarrow \infty}x_n > \displaystyle\lim_{n \rightarrow \infty}y_n$;
  • you can then find an $N$ such that $x_n > y_n$ for all $n > N$;
  • but then $x_n \le y_n$ can only hold for at most $N$ and thus finitely many $n$.
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  • $\begingroup$ $x_n>y_n$ for infinitely many $n$ doesn't contradict that $x_n\le y_n$ for infinitely many $n$. However, what you proved does suffice: $x_n>y_n$ for all $n\ge N$, so $x_n\le y_n$ for at most $N<\infty$ values of $n$, namely $n\in\{1,2,\dots,N\}$. $\endgroup$ – Cody Johnson May 30 '18 at 18:44
  • $\begingroup$ @CodyJohnson Of course, thanks! $\endgroup$ – StackTD May 31 '18 at 17:17
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The terms that satisfy the condition form a subsequence. Clearly the subsequential limit has lim x_n <= lim y_n . But this is just the limit of the sequences because the sequences converge.

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