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Consider two sequences $(a_n)$ and $(b_n)$ both contained in $\mathbb{R}$ and assume that these two sequences satisfy $|a_n - b_n| \rightarrow 0$, then this does imply that both $a_n$ and $b_n$ are convergent sequences? Or is it possible to have two divergent sequences but the absolute value of their difference forms a convergent sequence?

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Take $(a_n)=(b_n)=(1,-1,1,-1,...)$.

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Even more: if $(|a_n - b_n| \rightarrow 0)$ then ($a_n$ converges iff $b_n$ converges).

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  • $\begingroup$ So in words: if $|a_n - b_n| \rightarrow 0$ is satisfied, then either both $a_n$ and $b_n$ are convergent or both are divergent? $\endgroup$ – user124005 Jan 26 '14 at 18:45
  • $\begingroup$ Right! Because $|a_n - b_n| \rightarrow 0$ iff $a_n - b_n \rightarrow 0$ (only valid for zero!). $\endgroup$ – Martín-Blas Pérez Pinilla Jan 26 '14 at 18:47

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