# 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.

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.