This is question 30 in chapter 7 from royden
"Let $\{f_n\}$ be a sequence in C[a, b] and $\sum a_k$ a convergent series of positive numbers such that $|| f_{k+1}- f_k||_{\max} \leq a_k, \forall k$ Prove that $|f_{k+n}(x)- f_k(x) | \leq || f_{k+n}- f_k||_{\max} \leq \sum_{j=n}^{\infty}a_j ,\ \forall k,n $
Conclude that there is a function f$\in$ C[a, b] such that $\{fn\} \rightarrow f$ uniformly on [a, b] "
I know that, it is easy to solve this question if the series above starts from j=k , by adding and subtracting $f_{k+1},..., f_{k+n-1 } $ to the leftside and useing the triangular inequality.
But in this problem the seris starts from j=n. I tried hard to solve it., however I have not any good idea. Can you help me?? Any suggestion will be appreciated.