limit of infinite seqences

I am trying to prove this below proposition from real analysis royden pg 23 4th edition.

Let $\{a_n\}$ and $\{b_n\}$ be real sequences, if $a_n <= b_n$ then $\limsup\{a_n\} <= \liminf\{b_n\}$

but I have found a counter example to disprove this, please correct me what I understood wrong or what is wrong with the counter example or is there a typo in question.Let $\{b_n\} := \{4,2,4,2,4,2 ...\}$ and $\{a_n\} := \{3,1,3,1,3,1...\}$ then $\limsup\{a_n\} = 3 > 2 = \liminf\{b_n\}$