So let me put this together, does this all make sense, am I missing anything?
Suppose
and
are absolutely convergent series of real numbers. We need to show that
converges. We will show this using the comparison test.
The comparison test states: Suppose that we have two series,
and
, with
for all
and
for all
; then if
is convergent so is
.
Now by the definition of absolute convergence we know that:
for some real number
and
for some real number
.
Let us consider that
We show this is true:
Now consider that,
Since we have shown that every element is less than or equal to this new series, and it is clear that this new series converges since it is the sum of two convergent series multiplied by a constant. Hence by the comparison test, is convergent.