Let $\{a_n\}$ be a sequence. I understand $\limsup_{n \rightarrow \infty}{a_n}$ is the greatest value of convergent subsequences and $\liminf_{n \rightarrow \infty}{a_n}$ is the smallest value of convergent subsequences. For example $$a_n = \{1, 2, 3, 1, 2, 3, 1, 2, 3, ...\}$$ $a_n$ has subsequences that converge to $\{1, 2, 3\}$ so that $$\limsup_{n \rightarrow \infty}{a_n} = 3 \quad \liminf_{n \rightarrow \infty}{a_n} = 1$$
My trouble is understanding how to make that interpretation connect to the definitions $$\limsup_{n \rightarrow \infty}{a_n} = \inf_n{ \sup_{m\geq n}{a_m}} \\ \liminf_{n \rightarrow \infty}{a_n} = \sup_n{ \inf_{m\geq n}{a_m}}$$
Can someone try and connect the two ideas together for me to have a greater understanding of the definition? Thank you.
The text I'm using is Russell Gordon's Real Analysis, A First Course and Richard F. Bass' Real Analysis for Graduate Students.