How to prove $\limsup (x_{n}+y_{n})=\lim x_{n}+\limsup y_{n}$?

sequences-and-series calculus limsup-and-liminf

Solution 1:

Simply take a convergent subsequence $y_{n_k}$ such that $\limsup y_n=\lim y_{n_k}$. Then $\limsup(x_n+y_n) \geq \lim (x_{n_k}+y_{n_k})= \lim x_n+\limsup y_n$

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