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Let $(x_n)$ a real sequence such that $(x_n^2)$ is non-increasing and $(x_{n+1}-x_n)$ converges to $0$. Prove that $(x_n)$ converges.

My attempt:

$(x_n)^2$ is a decreasing sequence and $(x_n^2)\geq 0$ therefore the sequence converges to $l\geq 0$

If $l=0$ therefore $(x_n)$ converges to $0$.

I haven't managed to do the case $l\neq 0$, I tried with $\epsilon$ but I didn't succeed.

Thank you in advance for your help,

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    $\begingroup$ Your attempt doesn't make much sense. You need to prove that the sequence is Cauchy. $\endgroup$ – vadim123 Feb 18 '14 at 17:16
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The bounded and monoton sequence $|x_n|$ must converge to some limit $l$. Let $M=\{\,n\in\mathbb N\mid x_n>0, x_{n+1}\le 0\,\}$. If $M$ is finite, almost all $x_n$ have the same sign, which implies $x_n\to l$ or $x_n\to-l$ depending on that sign. If on the other hand $M$ is infinite, we have $$\lim_{M\ni n\to \infty} x_n=l $$ $$\lim_{M\ni n\to \infty} x_{n+1}=-l $$ $$ \lim_{M\ni n\to \infty} (x_{n+1}-x_n)=0 $$ hence $l=0$. But then $|x_n|\to 0$ implies $x_n\to 0$.

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  • $\begingroup$ I don't understand the difference between M finite and infinite :/ $\endgroup$ – user117932 Feb 20 '14 at 16:53
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If $x_n^2 \downarrow l>0$, then for every $n$ we have either $x_n\ge \sqrt{l}$ or $x_n \le -\sqrt{l}$. Now, from $x_{n+1}-x_{n} \to 0$ it follows that there exists $N$ such that for every $n>N$ $x_{n+1}$ and $x_n$ must have the same sign. Therefore all $x_n$ can be assumed positive for $n>N$. But in this case from $x^2_{n+1} \ge x_n^2$ it follows that $x_{n+1} \ge x_n$ $n> N$. Therefore $x_n$ is non increasing and bounded from below by $\sqrt{l}$. Therefore it is convergent. The same is the case if we assume all $x_n<-\sqrt{l}$ for $n>N.$

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  • $\begingroup$ Yes, I also used this fact. $\endgroup$ – kmitov Feb 19 '14 at 5:17
  • $\begingroup$ OK. You can think what you want. $\endgroup$ – kmitov Feb 19 '14 at 11:18
  • $\begingroup$ What is the difference between decreasing and non-increasing? $\endgroup$ – kmitov Feb 19 '14 at 11:34
  • $\begingroup$ Where in my answer is written that $x_n^2$ is decreasing? $\endgroup$ – kmitov Feb 19 '14 at 11:35
  • $\begingroup$ (The downvote it's not mine) Nowhere it's my point, in my exercice we assumed $(x_n)^2$ is decreasing (non-increasing) $\endgroup$ – user117932 Feb 19 '14 at 12:02

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