Absolute convergence implies that
$ \mid \mid s_m\mid −\mid s_n \mid \mid \space = \space \mid \mid x_{m}\mid +\mid x_{m−1}\mid +…\mid x_{n+1} \mid \mid \space \leq \space ϵ $
But $\mid s_m −s_n\mid \space = \space\mid x_{m} + x_{m-1} +… x_{n+1}\mid $
$ \because \space \mid x_{m} + x_{m−1} +… x_{n+1}\mid $ $\leq$ $ \mid \mid x_{m}\mid +\mid x_{m−1}\mid +…\mid x_{n+1} \mid \mid $ = $ \mid \mid s_m\mid −\mid s_n \mid \mid $
So $\space \mid s_m −s_n\mid \space \leq$ $ \mid \mid s_m\mid −\mid s_n \mid \mid $
Which is a contradiction. What is wrong in this ?