Sequences and Limit Superior

Suppose that xn x and the sequence (yn) is bounded. Show that

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lim (xn + yn) = lim xn + lim (yn).

I know that since (xn) converges lim xn = lim (xn) and that

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lim (xn + yn) </= lim xn + lim yn.

Thus the equality in this equation must come from the fact that (yn) is bounded, but I am not sure how to get there. Please help.

keywords: supremum

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