$\{ X_n \}$ is a sequence such that $\vert X_{n+1} - X_n \vert \lt \frac{1}{2^n}$ for all $n$. Then show that $\{ X_n \}$ is Cauchy.

Question

$\{ X_n \}$ is a sequence such that $\vert X_{n+1} - X_n \vert \lt \frac{1}{2^n}$ for all $n$. Then show that $\{ X_n \}$ is Cauchy.

I know that I want to show:

$\vert X_{m} - X_n \vert \lt \varepsilon$ for all $m,n \geq k$

I also know:

$m>n, |X_m - X_n| < 1/2^{m-1} + 1/2^{m-2} + \cdots+1/2^n$

Answer 1

Hint: $\frac{1}{2^{n-1}}=\frac{1}{2^{n}}+\frac{1}{2^{n+1}}+... $(Why?)

You can now bound your sum and find the desired $k$ without worrying about $m$.