If $(2x_{n+1}-x_n)$ converges to $x$, then show that $(x_n)$ converges to $x$.

Question

Let $(x_n)$ be a sequence in $\mathbb{R}$. Show that if $(2x_{n+1}-x_n)$ converges to $x$, then $(x_n)$ converges to $x$.

We don't know if the sequence $(x_n)$ is a convergent sequence or a cauchy sequence. If we are able to prove any one, then the problem is simple. Please provide any clue on how to go about.

Answer 1

Hint:

\begin{align} |x_m-x_n| =&\ \frac{1}{2}|2x_m-x_{m-1}+x_{m-1}-x_{n-1}+x_{n-1}-2x_n|\\ \leq&\ \frac{1}{2} |(2x_m-x_{m-1})-(2x_n-x_{n-1})|+\frac{1}{2}|x_{m-1}-x_{n-1}|. \end{align}

Probably a very bad hint!

Answer 2

Hint. Call $y_n = 2x_{n+1} - x_{n}$, let $N>0$ and look at $$\frac1{2^N} y_n + \frac1{2^{N-1}}y_{n+1} + \cdots + \frac12 y_{n+N-1} + y_{n+N}$$