Monotone convergence theorem for a decreasing recursive sequence from $n>n_0$

Question

Let a recursive sequence: $$a(0)=a\in\mathbb{R};\\ a(n+1)=f(a(n));\\$$ if the sequence is decreasing but not for all $n\in\mathbb{N}$, e.g $a(n+1)\leq a(n) \,\, \forall \, n\geq 2$, it holds the monotone convergence theorem? So I can say that exists the limit of the sequence (even if the decrease holds from $n=2$)?

Answer 1

It is required that the sequence is $\textit eventually$ monotonic, since it would still mean that $\textit eventually$ all the terms in the sequence are less than $\epsilon$ for every $\epsilon$. However, it is also required by the monotonic convergence theorem that the sequence is bounded. It doesn't look like you have given how $f$ is defined.