>>11062611You're confusing cardinals and sequences. Pretty weird error to make.
See a simple example:
Let u_n bet the sequel of the nonzero integers up until ten, and then only 0: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 0, 0, 0....
Let v_n be the sequel of prime intergers up until 7, and then only 0: 2, 3, 5, 7, 0, 0, 0....
v_n is a strict subsequence of u_n (for any n, there exist f(n)>=n such that v_n = u_f(n) and such that f is a stricly increasing function of the integers) and they both have the same limit, 0.
But the sum of u_n and sum of v_n are two infinite sums (though with only a finite number of nonzero terms). The sum of u_n has 55 has a limit (it's a stationary sequel). The sum of v_n has 17 as a limit (it's also a stationary sequel).
Another example (with no zero term) is with u_n = 1/n^2 and v_n = 1/(2^n)^2. Both converge to 0, the latter is a subsequence of the former, but the sum of the former is pi^2/6 which is an irrational and above 1.5 while the sum of the latter is 4/3 which is a rational and below 1.4.
