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If $x_n$ and $y_n$ are sequences such that $(x_n + y_n)$ is convergent. Does it mean that both $x_n$ and $y_n$ have to be convergent?

If it does not, please provide an example.

Thanks a lot!

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Pick your favorite divergent sequence. Call it $u_n$. Put $x_n = u_n$, $y_n = -u_n$.

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What if $x_n = (-1)^n$ and $y_n = (-1)^{n+1}$?

You can construct similar things using $\sin$ and $\cos$, for example.

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The sequence $x_n=n$ is divergent, and also $y_n=-n$. But $z_n=x_n+y_n=0$ is convergent. Then the answer is not, in general. The reciprocal, is of course true. If $x_n$ and $y_n$ are both convergent then any linear combination, in particular $(x_n+y_n)$ is convergent.

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