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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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