Let $0 \leq \alpha < 1$ and let f be a function from $\mathbb{R} $ into $\mathbb{R}$ which satisfies $ | f(x) - f(y)| \leq \alpha|x-y| \; \forall x,y \in \mathbb{R}.$ Let $a_{1} \in \mathbb{R}$ and let $a_{n+1} = f(a_{n})$ for $n=1,2....$ Prove that $\{a_{n}\} $is a Cauchy Sequence.
I have noticed that the sequence values are getting closer together but I haven't been able to show that they get arbitrarily close together. Also I know that if I can show that the sequence is bounded and monotone it is convergent and hence a Cauchy sequence by a theorem.