Does a bounded sequence with diminishing increment converge? In my research, I have the following discrete-time dynamical system
$M_k=\frac{1}{k+1} I_k+\frac{k}{k+1}M_{k-1},$
where $|I_k|<c$ is absolutely bounded, and $I_k$ depends on $M_{k-1}$.
Now, I want to prove that $M_k$ has a limit. I can rewrite the dynamic as follows
$M_k=\frac{1}{k+1} M_0+\frac{1}{k+1}\sum_{t=1}^k I_t.$
Since I know the initial condition $|M_0|<c$ is absolutely bounded, it means $|M_k|<c$ is also absolutely bounded. I can also rewrite the original dynamic as follows
$M_k-M_{k-1}=\frac{1}{k+1}(I_k-M_{k-1}),$
thus the increment $|M_k-M_{k-1}|<\frac{2c}{k+1}$ is diminishing.
How can I proceed from here to prove/disprove that $M_k$ has a limit?
Thank you in advance for any help.
Edit 1: After seeing the counter-examples some people constructed, I think it is better to illustrate a little bit more on the dynamic of $I_k$. In my modeling, $I_k$ is governed by a random process. Let $\omega_k$ denotes the realization of the state at time $k$ which is sampled $i.i.d.$ from a known distribution $\mu$. In addition to that, $I_k$ also depends on $M_{k-1}$. The simplified version of the dynamic of $I_k$ can be written as follows
$I_k=\alpha^{\omega_k} f(M_{k-1})+\beta^{\omega_k},$
where $\alpha^{\omega_k}, \beta^{\omega_k}$ are state-dependent parameters, $f(\cdot)$ is a real function and is Lipschitz continuous.
Will the additional information of $I_k$'s dynamic help to prove the existence of the limit of $M_k$? Maybe some variations of the Law of Large Numbers might work here?
 A: Not in general. Consider the sequence whose first terms are
$$
1,1+1/2,1+1/2+1/3, \dots 1+1/2+1/3+\dots 1/n
$$
where $n$ is chosen in such a way that $a_n\le 2$, $a_n+1/(n+1)>2$. Let the next group of terms be
$$
a_n-1/(n+1), a_n-1/(n+1)-1/(n+2),..a_n-1/(n+1)-1/(n+2)-\dots -1/m
$$
where $m$ is chosen in such a way that $a_{n+m}\ge -2$, $a_{m+n}-1/(m+1)<-2$
and so on. Then clearly
$$
|a_k-a_{k-1}|=1/k\to 0
$$
but your sequence oscillates between $-2$ and $+2$ without limit. Observe that this construction is possible because the harmonic series is divergent.
A: I'm afraid it's not true that $M_k$ will always have a limit. The sequence $I_k$ can be chosen in such a way that it won't. I'll take $c>1$ in this example, and $M_k=0$. Define $I_k$ such that
$I_0=1$, if $M_k \ge \frac{1}{2}$ then $I_k=-1$, if $M_k \le -\frac{1}{2}$ then $I_k = 1$ and if $-\frac{1}{2} < M_k < \frac{1}{2}$ then $I_k = I_{k-1}$.
This sequence will "attempt to converge" to $1$ in the sections where $M_k < \frac{1}{2}$ and to $-1$ in the sections where $M_k > -\frac{1}{2}$, therefore oscillating between $\frac{1}{2}$ and $-\frac{1}{2}$.
