# Integral proof

1. Feb 7, 2005

### quasar987

Consider f:[a,b] --> R, an integrable and bounded (ain't that implied by "integrable"?!) function and consider {a_n} a sequence that converges towards a and such that a < a_n < b (for all n). Show and rigorously justify that

$$\lim_{n \rightarrow \infty} \int_{a_n}^{b} f(x)dx = \int_{a}^{b} f(x)dx$$

All we found is the, imo, not very rigorous and seemingly too easy,

$$\int_{a}^{b} f(x)dx = \int_{a}^{a_n} f(x)dx + \int_{a_n}^{b} f(x)dx \Rightarrow \lim_{n \rightarrow \infty} \int_{a}^{b} f(x)dx = \lim_{n \rightarrow \infty} \int_{a}^{a_n} f(x)dx + \lim_{n \rightarrow \infty} \int_{a_n}^{b} f(x)dx$$
$$\Rightarrow \int_{a}^{b} f(x)dx = 0 + \lim_{n \rightarrow \infty}\int_{a_n}^{b} f(x)dx$$ qed

Does anyone with more insight see how to do this more rigorously or is this the way?

2. Feb 7, 2005

### learningphysics

Yes. If it's Riemann integrable, then it must be bounded. I don't know anything about Lebesgue integrals.

I'd say, use the second fundamental theorem of calculus to define an anti-derivative of f... call it F(x).

Then use the first fundamental theorem of calculus, to write your integral of f(x) from $$a_n$$ to b, as F(b)-F(an). Then the rest should be easy using the properties of limits.

Last edited: Feb 7, 2005
3. Feb 7, 2005

### quasar987

I think you're assuming that there exists a primitive. "Integrable" does not imply "there exist a primitive to f(x) such that the integral is F(b) - F(a)". Maybe f is discontinuous.

4. Feb 7, 2005

### learningphysics

Yes, you're right. Forget about the first fundamental theorem of calculus.

Your proof is right... except you need to show that:

$$lim_{n \rightarrow \infty} \int_{a}^{a_n} f(x)dx =0$$

According to the second fundamental theorem of calculus, we can define:

$$F(z)=\int_{a}^{z} f(x)dx$$

We don't need f(x) to be continuous to do this. So:

$$F(a_n)=\int_{a}^{a_n} f(x)dx$$

We can put this in your limit and the limit becomes:

$$lim_{n \rightarrow \infty} F(a_n)$$

But according to the second fundamental theorem of calculus F(z) is continuous. So:

$$lim_{n \rightarrow \infty} F(a_n)= F(lim_{n \rightarrow \infty}a_n)$$

which equals:

$$F(a)$$

and $$F(a)=\int_{a}^{a} f(x)dx =0$$

So that should do it.

Last edited: Feb 8, 2005
5. Feb 7, 2005

### quasar987

Very nice! Thank you for that.