Convergent Real Sequence/Examples/Difference between Adjacent Terms Bounded by alpha^n

Example of Convergent Real Sequence
Let $\alpha \in \R$ be a real number such that $0 < \alpha < 1$.

Let $\sequence {x_n}_{n \mathop \in \N_{>0} }$ be a sequence in $\R$ with the property:
 * $\size {x_{n + 1} - x_n} \le \alpha^n$

Then $\sequence {x_n}$ is a Cauchy sequence and hence converges.

Proof
Let $n > m$.

Then:

Let $\epsilon \in \R_{>0}$ be given.

Let $N$ be sufficiently large that:


 * $\dfrac {\alpha^m} {1 - \alpha} < \epsilon$

This is always possible, as shown by Sequence of Powers of Number less than One.

Hence the result, by Cauchy's Convergence Criterion.