Let be $f:[1,\infty)\to [0,\infty)$ monotonically decreasing and $\int\limits_1^{\infty}f(t)dt$ an improper integral. Further we can assume that $\int\limits_1^{\beta}f(t)dt\leq d$ for all $\beta\in[0,\infty)$, where $d\in\mathbb{R}$. We want to show that $\int\limits_1^{\infty}f(t)dt$ exists.
My approach:
We define a function $F(x):=\int\limits_1^{x}f(t)dt$. So we know that if $\lim\limits_{x\to\infty}F(x)$ exists then $\int\limits_1^{\infty}f(t)dt$ exists. We know from the theory of limits that $\lim\limits_{x\to\infty}F(x)$ exists iff for each sequence $\left(b_n\right)_{n\in\mathbb{N}}$ with $\lim\limits_{n\to\infty}b_n=\infty$ the sequence of the images $F(b_n)$ also converges, e.g. $\lim\limits_{n\to\infty}F(b_n)=c$.
As $f(x)\geq 0$ for all $x\in [1,\infty)$, I was wondering if we can simply argue that $F(b_n)=\int\limits_1^{b_n}f(t)dt$ is monotonically increasing. This would allow us to conclude that $F(b_n)$ is convergent (monotonicity and boundedness imply convergence of sequences, aka monotone convergence theorem) and hence $\int\limits_1^{\infty}f(t)dt$ exists.
However, I am not sure about this step because if the sequence $\left(b_n\right)_{n\in\mathbb{N}}$ is not monotonically increasing then it would not be guaranteed that $F(b_n)=\int\limits_1^{b_n}f(t)dt$ is monotonically increasing.
Am I right with my concerns? Or is it legit to argue with monotonicity and boundedness in this case?
EDIT
I think there is some ambiguity in the discussion in the comments below which causes a lot of confusion. I will clarify it a little bit. There exist at least two different versions of the monotone convergence theorem. One regards to sequences (see https://en.wikipedia.org/wiki/Monotone_convergence_theorem) and the other one to limits of functions (see for example: Proving a bounded monotone function has finite one-sided limits or Suppose that the function $f:I\rightarrow\mathbb{R}$ is monotonically increasing and bounded. Prove that the $\lim_{x\rightarrow a}f(x)$ exists). The expression $F(b_n)$ defines, as mentioned above, a sequence, precisely the sequence of images of $b_n$ under the function $F(x)$. As shown in my example it is wrong to say that $F(b_n)$ is increasing! So we cannot apply the monotone convergence theorem (which deals with sequences).
However, the monotone convergence theorem wich regards to limits of functions requires the function to be monotonically increasing and not the sequence $F(b_n)$. The function $F(x)$ is clearly monotonically increasing as $f(x)\geq 0$.
So I cannot apply the monotone convergence theorem of sequences but the monotone convergence theorem of limits of functions to show the existence of the limit.
