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

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)

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For sum reason this question was very hard to parse – user58512 Feb 14 '13 at 17:55
How can the sum of a sequence of positive numbers converge to zero? – Ross Millikan Feb 14 '13 at 18:05

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.

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What's going on today? Why the downvote? – 1015 Feb 14 '13 at 17:55
Why the downvote? – Hagen von Eitzen Feb 14 '13 at 17:57
+1 This is a great answer and great reasoning. – anon271828 Feb 14 '13 at 18:00
@anon271828 Thanks! But that's just the very beginning of a reasoning. – 1015 Feb 14 '13 at 18:02

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.

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