The following question is in regard to material sourced (and presumablty authored by) M.Ram Murty in the publication "Problems in Analytic Number Theory, Second Edition", page 18.
And I am starting to feel like the use of curly brackets is an inside joke amongst those further along the path than I am right now, but none the less I just woke up and I'm baffled by this first step, and really would like someone to explain this step in a generalized sense, from an axiomatic stance if possible.
The author declares the following to be very useful, for which it is indeed very useful for me at this point:
Suppose ${\{a_n}\}_{n=1 }^\infty$ is a sequence of complex numbers and $f(t)$ is a continuously differentiable function on $[1,x]$.
Set $$A(t)=\sum_{n \leq t}a_n$$ Then $$\sum_{n \leq x}a_n f(n)=A(x)f(x)-\int _{x}^{1}A(t)f^{'}(t)dt$$ First we suppose $x$ to be a natural number, we write the left hand side as: $$\sum_{n \leq x}a_n f(n)=\sum_{n \leq x}{\{A(n)-A(n-1)}\}f(n)=\sum_{n \leq x}A(n)f(n)-\sum_{n \leq x-1}A(n)f(n+1)$$
Many thanks in advance.