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Let $x_n$ and $y_n$ be two bounded sequences. Assume that $y_n$ converges to some value b. I seek to prove that lim sup($x_n$ + $y_n$) = lim sup($x_n$ ) + b. I did prove that lim sup($x_n$ + $y_n$) $\leq$ lim sup($x_n$ ) + lim sup($y_n$) for any two bounded sequences, but I'm not sure how to continue, and not sure if this helps in the first place.

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    $\begingroup$ Please see revision of title to understand how to encode mathematics on this site. $\endgroup$ – Did Sep 24 '18 at 18:47

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