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For any real $a_1$ define $a_{n+1}=\cos(a_n)$, how to prove that the limit of this sequence exist for all real $a_1$?

Context: I was solving this question : Study convergence of $a_n$, where $a_1=1$, $a_{n+1}=\cos(a_n)$.

I am sure that this sequence converges to the solution of $x=\cos(x)$ which is greater than $0$ so the answer is that the series diverge, I am sure there are better ways to prove the divergence than proving that the sequence $a_n$ converge but I found the other question more intersting

I couldn't prove that the limit exist after an hour of trying I didn't reach anything useful so I decided to ask here, I also want to ask if the limit will always exist if $a_1 \in \mathbb{C}$? This question , unlike the real version, is more complicated since $\cos$ is not bounded in $\mathbb{C}$.

Red Five
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pie
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  • See also duplicates here, here, here, etc. – Chris Grossack Jul 25 '24 at 01:29
  • @ChrisGrossack This only answers the first question about reals. – pie Jul 25 '24 at 01:29
  • @Gonçalo The only missing part is the answer for " I also want to ask if the limit will always exist if $a_1\in C$ ?" I think this question would get closed without an answer, should I post another question dedicated to the case when $a_1\in \mathbb{C}$?? – pie Jul 25 '24 at 01:41
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    It's probably a good idea to post another question asking about the $\mathbb{C}$ case explicitly. You can reference the fact that the duplicates I pointed out don't cover this case, and this question is quite likely to be closed because in the title and first few paragraphs you only ask about the real value $a_1$. Lots of people (myself included, apparently) don't read these sorts of "obvious duplicates" very closely before voting to close. – Chris Grossack Jul 25 '24 at 01:46

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