How to get the last step?

Question

I try to interchange the integral and limit $$\int_{c}^{\infty}f(y)dy=\lim_{k\to\infty}\int_{c}^{k}f(y)dy$$ why the last step holds? Why claims the last step using the Monotone convergence theorem?

Answer 1

Define $f_n(y)=\frac{1}{y^2+1}1_{[c,k]}(y)$

Note that $1_{[c,k]}(y) \to^{k \to +\infty}1_{[c,+\infty)}(y)$ pointwise.

and $1_{[c,k]} \leq 1_{[c,k+1]}$ so you can use the monotone convergence theorem since $\frac{1}{y^2+1} \geq 0$

This is to answer the previous question,before you edited and changed it.