Finding a limit of a sequence or proving it diverges

[nikolafmf]

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.

Homework Equations

?

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]

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.

Homework Equations

?

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]

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]

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.

 

[nikolafmf]

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 :)