# Finding a limit of a sequence or proving it diverges

## Homework Statement

Given is a sequence: sin(1), cos(sin(1)), sin(cos(sin(1))) etc. Find the limit of the sequence or prove it diverges.

?

## The Attempt at a Solution

One way to prove a sequence diverges is to find two subsequences which converge to different limits, but I could not find such.

I would be thankful for any idea :)

Dick
Science Advisor
Homework Helper

## Homework Statement

Given is a sequence: sin(1), cos(sin(1)), sin(cos(sin(1))) etc. Find the limit of the sequence or prove it diverges.

?

## The Attempt at a Solution

One way to prove a sequence diverges is to find two subsequences which converge to different limits, but I could not find such.

I would be thankful for any idea :)

If there is a limit L, then it must satisfy both sin(L)=L and cos(L)=L, mustn't it?

haruspex
Science Advisor
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Gold Member
2020 Award
Suppose it does converge to some value. What equations could you deduce regarding that value?
[Dick beat me to the Submit, and was a little more generous with the hint.]

Dick
Science Advisor
Homework Helper
Suppose it does converge to some value. What equations could you deduce regarding that value?
[Dick beat me to the Submit, and was a little more generous with the hint.]

Yeah, probably too generous in retrospect. I like yours better as a starter hint.

Last edited:
Yes, it is true, it must satisfy both sin(L)=L and cos(L)=L, from which follows that the sequence diverges :)

Thank you very much to both for the help :)