Question: Consider the set of functions L such that the functions are Lipschitz. That is: $$|f(x)- f(y)| \leq K|x-y|$$. Consider the metric $$\rho (f_1,f_2) = \sum_{j=1}^\infty 2^{-j} \text{sup}_{x \in [-j,j]} |f_1(x)-f_2(x)|$$ Show that $L$ is a complete metric space with metric $\rho$.
Attempt: Assume a Cauchy sequence. Then $$\rho(f_n,f_m)< \epsilon$$ This means $$|f_n(x)-f_m(x)|\leq \epsilon$$ Thus its Uniformly Cauchy and $f_n \rightarrow f$ uniformly and we are done.
Is this correct? I am not sure about the step where $\sum_{j=1}^\infty 2^{-j} \text{sup}_{x \in [-j,j]} |f_1(x)-f_2(x)|\leq \epsilon \implies |f_n(x)-f_m(x)|\leq \epsilon$