$\textbf{The Problem:}$ Suppose $f$ is Riemann integrable on $[a,b]$ for every $b>a$ where $a$ is fixed. Define $$\int_{a}^{\infty}f(x)dx=\lim\limits_{b\to\infty}\int_{a}^{b}f(x)dx$$ if the limit exists and is finite. In that case, we say that the integral on the left converges. If it also converges after $f$ has been replaced by $\vert f\vert$, it is said to converge absolutely.
Assume $f\geq0$ and that $f$ decreases monotonically on $[1,\infty)$. Prove that $$\int_{1}^{\infty}f(x)dx<\infty\quad\text{if and only if}\quad\sum^{\infty}_{n=1}f(n)<\infty.$$
$\textbf{My Attempt:}$ Let $N\in\mathbb N$ with $N>1$ and consider the partition of the interval $[0,N]$ given by $\mathcal P=\{1,2,\dots,N-1,N\}.$ Since $f$ decreases monotonically on $[1,\infty)$ we see that the upper and lower sums are equal to, respectively, $$\mathcal U(f,\mathcal P)=\sum^{N-1}_{n=1}f(n),$$ $$\mathcal L(f,\mathcal P)=\sum^{N}_{n=2}f(n).$$ Then since $$\sum^{N}_{n=2}f(n)\leq\int_{1}^{N}f(x)dx\leq\sum^{N-1}_{n=1}f(n),$$ and $N>1$ was arbitrary, we see that $$\int_{1}^{\infty}f(x)dx<\infty\quad\text{if and only if}\quad\sum^{\infty}_{n=1}f(n)<\infty.$$
Do you agree with my proof above? If not, please tell me, and even if you do but think it could sue improvement, please tell me.
Thank you for your time and appreciate any advice.
Sorry for the fact that I don't have enough reputations to add a comment.
Your proof is OK. But if you are a beginner on this, I recommend you to write more precisely.
I mean, for example, you should ask yourself: To prove the sufficiency, is it enough if you just consider the integral from 1 to $N$ where $N\in\mathbb N$? Notice that in the definition, $b$ is arbitrary.