Is this statement True or False? This question is within subseqence and $\limsup$.
$$\bf\text{If $(s_n)$ is unbounded above, then $(s_n)$ contains a subsequence that has $+ \infty$ as a limit.}$$
I say true. Because if $(s_n)$ is unbounded above, then the limit is $+ \infty$. So, at least one subsequence goes to $+ \infty$.
But the solution key says: "False. Let $(s_n) = (1, 0, 1, 0, 1,...)$. Then $\limsup s_n = 1$, but for $\epsilon = 1/2 $ there are infinitely many terms smaller than 1 - 1/2."
I don't understand the solution. I saw a similar question here, which says it is True. Is the solution wrong?