<s>[INST] How to show that this sequence is monotonic We are given that $(s_{n})$ is a bounded sequence. My goal is to show that $(s_{n})$ is monotonic, and thus it converges. It has the following property  $$s_{n+1} \geq s_{n} - \frac{1}{2^n}$$
I have tried induction, because intuitively the sequence monotonically increasing, but it fails at the basis step but works on the induction step.  Any ideas? 
 [/INST] $s_n$ is not necessarily monotonic given the information provided.
Example.
Let $s_n = \frac{2}{3} \left(\frac{-1}{2}\right)^n $.
Then $s_n$ is bounded, and
\begin{align*}
s_{n+1} - s_n
&= \frac{2}{3}\left(\frac{-1}{2}\right)^{n+1} - \frac{2}{3}\left(\frac{-1}{2}\right)^n \\
&= \frac{2}{3}\left(\frac{-1}{2}\right)^n \left( \frac{-1}{2} - 1\right) \\
&= -\left(\frac{-1}{2}\right)^n \\
&= (-1)^{n+1} \left(\frac{1}{2}\right)^n \ge \frac{-1}{2^n}. \\
\end{align*}
However, $s_n$ is clearly not monotonic.
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