The following is an infinite product, with $a_n$ real, not complex.
$$ P= \prod (1+a_n)$$
A common step in deriving convergence criteria is to take the logarithm,
$$\ln(P) = \sum \ln(1+a_n)$$
and then use $1+x\leq e^x$ to establish
$$ \ln(P) \leq \sum a_n $$
This tells us that if the sum is bounded, then so is the product $P$.
If we then assert that $a_n>0$, then the sum grows monotonically (no oscillation), and so the boundedness is actually convergence.
Question: It is commonly stated that if the sum $\sum a_n$, with $a_n>0$, converges then so does the product. Why?
Is it because the partial products increase monotonically too?
For all $n$, $a_n > 0$ and, hence,
$$P_{n+1} = P_n(1+a_{n+1}) > P_n$$
The sequence of partial products is both bounded and increasing ...