Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 3, Problem 5

Problem

For any two real sequences $\{a_n\}$, $\{b_n\}$, prove that

$$ \limsup_{n \rightarrow \infty} (a_n + b_n) \leq \limsup_{n \rightarrow \infty} a_n + \limsup_{n \rightarrow \infty} b_n $$

Answer

Note that there exists $n_k$ such that $a_{n_k} + b_{n_k} \rightarrow \limsup_{n \rightarrow \infty} (a_n + b_n)$.

$\limsup_{n \rightarrow \infty} (a_n + b_n) = \lim_{k \rightarrow \infty} (a_{n_k} + b_{n_k}) \leq \lim_{k \rightarrow \infty} (\sup_{m \geq k}a_{n_m} + \sup_{m \geq k}b_{n_m}) = \lim_{k \rightarrow \infty}\sup_{m \geq k}a_{n_m} + \lim_{k \rightarrow \infty}\sup_{m \geq k}b_{n_m} = \limsup_{n \rightarrow \infty} a_n + \limsup_{n \rightarrow \infty} b_n$.

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