Show Convergence of a Series Involving Functions that Satisfy Lipschitz.

Question

Let $K \geq 0$ and $L:=\{f:R \rightarrow R\}$ such that $|f(x)-f(y)| \leq K|x-y|$. Consider $\rho(f_1,f_2) = \sum_{j=1}^\infty 2^{-j}\text{sup}_{x \in [-j,j]} |f_1(x)-f_2(x)|$.

Show that $\rho$ converges.

Attempt: I was trying to apply the M test.

So I was trying to bound $$|2^{-j}\text{sup}_{x \in [-j,j]} |f_1(x)-f_2(x)||$$ but I am stuck.

I know that difference of two Lipschitz continuous functions is Lipschitz continuous but even that doesn't help me.

Answer 1

For $x \in [-j,j]$ we have $|f_1(x)|\leq |f_1 (0)|+K|x| \leq |f_1 (0)|+ Kj$. Similarly $|f_2(x)| \leq |f_2(0)|+Kj$. Hence $|2^{-j} \sup_{x \in [-j,j]} |f_1(x)-f_2(x)|| \leq 2^{-j} (|f_1 (0)|+|f_2 (0)|+2Kj)$. The three series $\sum |f_1(0)|2^{-j}$,$\sum |f_1(0)|2^{-j}$ and $\sum (2Kj)2^{-j}$ are all convergent so M-test can be applied.