Let $(x_n)$ be a bounded sequence such that $x_{n+1} \geqslant x_n - \frac{1}{2^{n}}$ for every $n \in \mathbb{N}$, show that $(x_n)$ converges.

Question

Let $(x_n)$ be a bounded sequence such that $x_{n+1} \geqslant x_n - \frac{1}{2^{n}}$ for every $n \in \mathbb{N}$, show that $(x_n)$ converges.

I know that if I can prove that $(x_n)$ is monotone, then by the monotone convergence theorem, $(x_n)$ would converge, but I don't know where to start to show that. There was a suggestion saying that I should first prove that $x_n - \frac{1}{2^{n}}$ is monotonically increasing, but that leads to a contradiction.

Answer 1

Put $$y_n:=x_n-{1\over 2^{n-1}}\leq\sup_k x_k=\xi<\infty\ .$$ Then $$y_{n+1}=x_{n+1}-{1\over 2^n}\geq x_n-{2\over 2^n}=y_n\qquad(n\geq1)\ .$$ It follows that the $y_n$ form a bounded increasing sequence, hence converge to a real number $\eta$. This allows to conclude that $$x_n=y_n+{1\over 2^{n-1}}\to\eta\qquad(n\to\infty)\ .$$