Convergence of improper integrals theorems

[R_beta.v3]

Homework Statement

I'm trying to prove these two theorems
a) if $0 \leq f(x) \leq g(x)$ for all x $\geq 0$ and $\int_0^\infty g$ converges, then $\int_0^\infty f$ converges
b) if $\int_0^\infty |f|$ converges then $\int_0^\infty f$ converges.

Obviously assuming f is integrable on every interval [0, N], N $\geq 0$.

Homework Equations

The Attempt at a Solution

For a).
Let F be defined by $F(x) = \int_0^x f$. $F$ is bounded above by $\int_0^\infty g$. and since, $0 \leq f(x)$ for all x, F is non-decreasing. So $\displaystyle\lim_{x\rightarrow \infty} F(x)$ exists.

For b)
$0 \leq f(x) + |f(x)| \leq 2|f(x)|$ for all x. and $\int_0^\infty 2|f|$ is convergent, so $\int_0^\infty (f + |f|)$ is convergent by (a).
The existence of $\displaystyle\lim_{x\rightarrow \infty} \int_0^x |f|$ and the existence of $\displaystyle\lim_{x\rightarrow \infty} \int_0^x (f + |f|) = \displaystyle\lim_{x\rightarrow \infty}\left( \int_0^x f + \int_0^x |f| \right)$. implies the existence of $\displaystyle\lim_{x\rightarrow \infty} \int_0^x f$

 

[Ray Vickson]

Homework Statement

I'm trying to prove these two theorems
a) if $0 \leq f(x) \leq g(x)$ for all x $\geq 0$ and $\int_0^\infty g$ converges, then $\int_0^\infty f$ converges
b) if $\int_0^\infty |f|$ converges then $\int_0^\infty f$ converges.

Obviously assuming f is integrable on every interval [0, N], N $\geq 0$.

Homework Equations

The Attempt at a Solution

For a).
Let F be defined by $F(x) = \int_0^x f$. $F$ is bounded above by $\int_0^\infty g$. and since, $0 \leq f(x)$ for all x, F is non-decreasing. So $\displaystyle\lim_{x\rightarrow \infty} F(x)$ exists.

For b)
$0 \leq f(x) + |f(x)| \leq 2|f(x)|$ for all x. and $\int_0^\infty 2|f|$ is convergent, so $\int_0^\infty (f + |f|)$ is convergent by (a).
The existence of $\displaystyle\lim_{x\rightarrow \infty} \int_0^x |f|$ and the existence of $\displaystyle\lim_{x\rightarrow \infty} \int_0^x (f + |f|) = \displaystyle\lim_{x\rightarrow \infty}\left( \int_0^x f + \int_0^x |f| \right)$. implies the existence of $\displaystyle\lim_{x\rightarrow \infty} \int_0^x f$

So, what is your question?

 

[R_beta.v3]

Sorry. I just want to make sure that my proof is correct? I'm studying by myself.