Riemann Integral - little proof help

[irresistible]

Homework Statement

Suppose f is integrable for all x in[a,b] and f(x)>C ( C is some constant),
Must show that 1/f is also integrable.

Homework Equations

f is integrable implies Upf-Lpf<\epsilon for some partition in [a,b]

The Attempt at a Solution

Therefore, I must come up with a good \epsilon such that
Lp(1/f) - Up(1/f) <\epsilon

Also f is bounded because it's integrable so there must be some m,M such that
f([a,b])= [m,M]
in other words f acheives it's minimum and maximum points on the interval.

 

[Billy Bob]

Some hints. For simplicity let's just look at one subinterval I=[x_i,x_{i+1}]

On I, we have m\le f(x)\le M (although really there is no guarantee that m and M are achieved, we still know f is bounded as you said).

Where does C fit in this inequality?

Take reciprocals of the inequalities 1/M\le 1/f(x)\le 1/m and put 1/c in the correct place, and furthermore where is 0?

Suppose M-m<delta.

Then \frac1m-\frac1M=\frac{M-m}{Mm} and you can find an upper bound for the last fraction in terms of delta and C.