Euler's summation formula proof

Question

The following proof is from Apostol's book:

Questions:

1. On the first line of the proof, he uses '{}' just as brackets or do they have other meaning like $[x]$ being the floor function?

2. right before equation (6), why does the summation from $m+1$ up to $k$ become $kf(k)-mf(m)$?

3. at equation (6) when he substitutes back $x,y$ i'm not sure why are the two integrals equal?

Answer 1

1. They are simply brackets, no specific meanings.
2. Try to write all of them. Then the terms cancelled like this: $$ 3f(3) - 2f(2) + 2f(2) - 1f(1) = 3f(3) - 1f(1). $$
3. Since $$ \int_k^x \lfloor t \rfloor f'(t) \mathrm d t = k \int_k^x f'(t) \mathrm d t = f(k) - f(x). $$ Same for the other term.

Answer 2

The proof in the question above has some mistakes. This is the newer one: