Given sequences $(x_n)$, convergent, but $(y_n)$ is divergent, then $(x_n + y_n)$ is divergent.

I am confident that it is true, but having trouble getting the formalities correct. I have tried proof by contradiction, i.e. assuming $\forall \varepsilon > 0 \; \exists N \in \mathbf{N}:$ $$ |(x_n + y_n) - L| < \varepsilon \qquad \forall n > N.$$ This seems to lead me nowhere. Equivalently, trying to find an $\varepsilon$ such that for all $N$ $|(x_n + y_n) - L| \geq \varepsilon$ have not gotten me any further. Any hints are appreciated.

  • 2
    $\begingroup$ Do you know that the sum (and difference) of two convergent sequences is convergent? $\endgroup$ – David Mitra Jun 13 '16 at 9:30
  • $\begingroup$ Yes that I have proven. $\endgroup$ – user305860 Jun 13 '16 at 9:33

If $(a_n)$ and $(b_n)$ are convergent, then so is $(a_n+b_n)$ and $(ca_n)$ for any scalar $c$. Now, assume that $(x_n)$ and $(x_n+y_n)$ are convergent to deduce that $(x_n+y_n-x_n)=(y_n)$ is convergent.

  • $\begingroup$ Well, that was easy.. $\endgroup$ – user305860 Jun 13 '16 at 9:34
  • $\begingroup$ Most of the time there is a very elegant solution for problems if you use some of the results you have proven! $\endgroup$ – sqtrat Jun 13 '16 at 9:35

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