Sum of two absolutely convergent series is also absolutely convergent?

r_n = ∑[0 to n] a_n
r = ∑[0 to infinity] a_n

the r_n is convergent
for any epsilon > 0 there exists an N such that for all n>N |r_n - r| < epsilon

suppose s_n as also a convergent series.
now lest look at r_n + s_n

you need to show that.

for any epsilon > 0 there exists an N such that for all n>N |r_n+s_n - (r+s)| < epsilon


Now you have shown that you get get |r_n - r| and |s_n-s| to be arbitrarily small.

epslion_2 = 1/2 epslion

for any epsilon > 0 there exists an N such that for all n>N |r_n - r| < epsilon_2
for any epsilon > 0 there exists an N such that for all n>N |s_n - s| < epsilon_2

|r_n+s_n - (r+s)| < or = |r_n - r| + |s_n - s| by the triangle inequality
if n > N
|r_n+s_n - (r+s)| < 2 epsilon_2
|r_n+s_n - (r+s)| < epsilon

The proof is in the link:

https://proofwiki.org/wiki/Sum_of_Absolutely_Convergent_Series