Absolute value of a function integrable?

[tomboi03]

this is the question,
Prove that if f is continuous on (a,b] and if |f| is bounded on [a,b] then f is integrable on [a,b]. (note: it is not assumed that f is continuous at a.)

I know you have to use the upper and lower bounds to prove this statement but i don't know where to start?

Thanks

 

[lurflurf]

let eps>0
H(x):={y!=x| |f(x)-f(y)|<eps/(b-a)}
Union[H(x)|x in [a,b]]
is an open cover (by continuity of f) of [a,b] a compact set so we may chose a finite subcover
is P is any partition at least as fine as the open cover will have
U-L<eps
qed