If $\displaystyle (a_{n_k})$ is a convergent subsequence of $\displaystyle (a_n)$, show that lim inf $\displaystyle _{n--> infinity}$ $\displaystyle a_n$ $\displaystyle <= $$\displaystyle lim_{k-->infinity}$$\displaystyle a_{n_k}$ $\displaystyle <=$ lim sup$\displaystyle _{n-->infinity}$$\displaystyle a_n$.

Im not sure what lim inf $\displaystyle _{n--> infinity}$ $\displaystyle a_n$ [tex] and lim sup$\displaystyle _{n-->infinity}$$\displaystyle a_n$ mean, since $\displaystyle inf a_n$ and $\displaystyle sup a_n$ are a number