Proving Limits of Infinite Integrals with LaTeX

[mscbuck]

Homework Statement

Prove that if $$\int_{-\infty}^{+\infty} f$$ exists, then $$\lim_{N\rightarrow \infty$$ of $${\int_{-N}^{N} f}$$ exists and is equal to the first equation.

Show moreover, that $$\lim_{N\rightarrow \infty$$ of $${\int_{-N}^{N+1} f}$$ and $$\lim_{N\rightarrow \infty$$ of $${\int_{-N^2}^{N} f}$$ both existThe attempt at a solution

It's taking me a really long time to write this out in LaTeX and it honestly looks worse than words when I finish it because I'm trying to learn it, so for now I"ll type in words.
My first step was split up the integrals. I have:

| Integral from 0->Inf of f MINUS Integral from 0->M of f | < 1/2E , and the correlating one for -Infinity to 0.

I then assumed an h(n) > M and a g(N) < -M for all N to come up with a generalization, and set up a large inequality and from that I believe by the triangle inequality I prove my result. But I"m unsure so if hopefully someone can check, that'd be great!

Thanks!

 

[ystael]

Because this is an area where different authors choose to set up their definitions in widely differing ways, it would be helpful if you gave your precise definition for $$\int_{-\infty}^{+\infty} f$$ (and for $$\int_0^\infty f$$ if that's involved in the previous definition). Also indicate whether this is the Riemann or Lebesgue integral. That way we can avoid leading you down a path that doesn't work with your exact set of assumptions.