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I am trying to come up with different examples of the property that

$\limsup_{n\to\infty} (s_n + t_n) \leq \limsup_{n\to \infty} s_n+\limsup_{n\to \infty} t_n$

So I thought of

$s_n = \left \{-3,1,-1,1,-1,1,-1,...\right \}$ and $t_n = \left \{0,-1,1,-1,1,-1,1,...\right \}$

So $(s_n + t_n) = \left \{-3,0,0,0,0,0,... \right \}$ and $\limsup_{n\to\infty} (s_n + t_n)= 0$ and $\limsup_{n\to\infty} s_n = 1$ and $\limsup_{n\to\infty} t_n = 1$

Is this correct?

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@GerryMyerson, yes. Oh shoot I made a typo – Hawk Oct 8 '12 at 2:32
up vote 3 down vote accepted

Yes. This is an example showing that the inequality can be strict.

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