:Prove BV function bounded and integrable

[kfdleb]

Homework Statement

f is of bounded variation on [a;b] if there exist a number K such that

$$\sum$$$$^{n}_{k=1}$$|f(a_k)-f(a_k-1)| $$\leq$$K

a=a_0<a_1<...<a_n=b; the smallest K is the total variation of f

I need to prove that
2) if f is of bounded variation on [a;b], then it is integrable on [a;b]

2. The attempt at a solution

i thought of using triangle inequation such that
0<=|f(b)-f(a)|<=$$\sum$$$$^{n}_{k=1}$$|f(a_k)-f(a_k-1)| $$\leq$$K

i did prove that f is bounded by contradiction that K is real, not infinite, nevertheless I'm not sure abt proving integrability
any help is really appreciated
thanks

 

[rochfor1]

You have a bounded function on a finite measure space...