A stupid one... (just to be sure)
If I have a series that is converge absolutely so the series is converges?
(it is implies from Cauchy criterion?)
Actually it is.
Note that for any infinite series , it can never be any greater than the sum of its absolute values.
Now simply apply the comparison test.
Note though that this only half proves the theorem, however you can also bound the series below...
So we can say
and since the series is bounded by the sum of the absolute values, that means if the sum of the absolute values converges, then the series is also convergent.