Limsup along even indices

Let $f:\mathbb{N}\to\{1,2,3,…,100\}$ be an onto function. Consider the following statements. $P$: If $g(n)=\sup\{f(2n),f(2n+2),f(2n+4),…\}$, $n\ge 1$, then $\lim_{n\to\infty}g(n)=\limsup_{n\to\infty}f(n)$. $Q$: If $h(n)=\inf\{f(n),f(n+1),f(n+2),…\}$, $n\ge 1$, then $\lim_{n\to\infty}h(4n+1)=\liminf_{n\to\infty}f(n)$. Then