If the sum converges to zero, does that mean that each sequence converges to zero?

Question

If we are given that the sequence sum of two sequences of positive real numbers converges to zero, does that mean that each sequence converges to zero? (by the squeeze theorem)

Answer 1

Let $x_n$ and $y_n$ be your sequences.

The answer is yes since $$ 0\leq x_n\leq x_n+y_n\quad\mbox{and}\quad 0\leq y_n\leq x_n+y_n $$ and then you can apply the squeeze theorem.

Answer 2

If the sequence $a_n+b_n$ converges to zero and $a_n, b_n$ are positive then $a_n$ (and by symmetry $b_n$) converges to zero because it's bounded below by 0 and above by some sequence that tends towards zero.