Absolute value of the difference between two sequences

Question

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?

Answer 1

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

Answer 2

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