How to show that the"$\le$" in $\lim\sup(s_t+t_n)\le\lim\sup s_n+\lim\sup t_n$ cannot be replaced by $=$?

Question

How to show that the"$\le$" in $\lim\sup(s_t+t_n)\le\lim\sup s_n+\lim\sup t_n$ cannot be replaced by $=$?

I think this means we need to show an example that $\lim\sup(s_t+t_n)\lt\lim\sup s_n+\lim\sup t_n$. However, I couldn't think of one. Could someone suggest a valid example

Answer 1

Let $s_n$ be the sequence $0,1,0,1,0,1,0,1,\ldots$

Let $t_n$ be the sequence $1,0,1,0,1,0,1,0,\ldots$

Answer 2

Choose for example $s_n = (-1)^n$, $t_n = (-1)^{n+1}$. You clearly have $s_n + t_n = 0$ for every $n$, hence $\limsup (s_n + t_n) = 0$, but $\limsup s_n = \limsup t_n = 1$.