Proving the Limit of Integral over f:[a,b] to R

[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?

 

[learningphysics]

Consider f:[a,b] --> R, an integrable and bounded (ain't that implied by "integrable"?!)

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

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

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.

 

[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.

 

[learningphysics]

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.

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.

 

[quasar987]

Very nice! Thank you for that.