The commonly used methods to judge whether a sequence has a limit are: definition method, Cauchy convergence method, pinch method, simplification method, reflexive reference method, monotone bounded method and so on. This problem can only be solved by monotone bounded method, so the key is to judge the monotonicity of {an}!

Prove:

Constructor:

F(x)=x-sinx, where x≥0.

Derivation:

f'(x)= 1-cosx≥0

∴f(x) monotonically increases in its domain.

And:

f(0)=0

∴x-sinx≥0

Namely:

X≥sinx, where: x≥0

Therefore:

a(n+ 1)=sinan