How can we show that the following sequence converges?

$(a_n)$ is a bounded sequence with the following condition

$a_{n+1}\geq a_n-\frac{1}{2^n}$

The sequence converges, but how do we show it?

Let $$ s_n = \sum_{i=1}^{n} \frac{1}{2^i} $$ Note that $$ s=\sum_{n=1}^{\infty} \frac{1}{2^n} < \infty $$ Consider $\tilde{a_n} = a_n + s_{n-1}$. We have that $(\tilde{a}_n)$ is bounded. Also, we have $$ \tilde{a}_{n+1} = a_{n+1} +s_{n} \ge a_n - \frac{1}{2^n} + s_{n} = a_n + s_{n-1} = \tilde{a}_n $$ so that $(\tilde{a_n})$ is increasing. By the monotone convergence theorem $(\tilde{a_n})$ converges which implies that $(a_n)$ converges.