Let $\displaystyle f$ be a function defined on all of $\displaystyle \mathbb{R}$, and assume there is a constant $\displaystyle c$ such that $\displaystyle 0<c<1$ and $\displaystyle |f(x)-f(y)|\leq\\k|x-y|$ for all $\displaystyle x,y\in\mathbb{R}$.

a) Show $\displaystyle f$ is continuous on $\displaystyle \mathbb{R}$.

Let $\displaystyle c$ be a real number and $\displaystyle \epsilon>0$. Then $\displaystyle |f(x)-f(c)|\leq\\k|x-c|$ by definition. So, let $\displaystyle \delta=\frac{\epsilon}{k}$. It follows that $\displaystyle |x-c|<\delta$ implies $\displaystyle |f(x)-f(c)|<k\frac{\epsilon}{k}=\epsilon$.

b) Pick some point $\displaystyle y_{1}\in\mathbb{R}$ and construct the sequence

$\displaystyle (y_{1},f(y_{1}),f(f(y_{1})),...)$.

In general, if $\displaystyle y_{n+1}=f(y_{n})$, show that the resulting sequence $\displaystyle (y_{n})$ is a Cauchy sequence. Hence, we may let $\displaystyle y=limy_{n}$.

I am not sure how to approach this one. Some help getting started would be appreciated. It seems almost trivial, but I am not sure how formalize it.