1. Jan 17, 2010

### kfdleb

1. The problem statement, all variables and given/known data

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

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

Last edited: Jan 17, 2010
2. Jan 17, 2010

### rochfor1

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