I can't understand this proof from my old lecture notes. $\liminf$ is defined as: \begin{align*} \liminf_{z\to\infty}f(z) = \inf_{x< y} \sup_{y< z}f(z) \end{align*} and $\limsup$ is defined similarly. We have already shown that if $f$ is an increasing (decreasing) function then $\sup f$ ($\inf f$) is unique, if it exists, hence the definition of liminf is safe. We have also shown that \begin{align*} \inf_{y<z} f(z) \leq \sup_{y<z} f(z). \end{align*} We assume that both $\liminf f$ and $\limsup f$ exist. The proof is presented as follows: \begin{align*} \inf_{y<z} f(z) &\leq \sup_{y<z} f(z)\\ \sup_{x<y} \inf_{y<z} f(z) &\leq \sup_{x<y} \sup_{y<z} f(z)\\ \inf_{x<y} \inf_{y<z} f(z) &\leq \inf_{x<y} \sup_{y<z} f(z) \end{align*} I don't see how this proves the theorem. Could somebody please explain the conclusion?
P.S. At this stage we do not have $\lim$ in our toolbox.
The correct definitions should be $$\liminf_{z\to\infty} f(z)=\sup_y\inf_{z\ge y}f(z),\qquad\limsup_{z\to\infty} f(z)=\inf_{y}\sup_{z\ge y}f(z). $$ Here is two-line proof: $\;\inf_{z\ge y} f(z) \leq \sup_{z\ge y} f(z)$, hence \begin{align*} \inf_{z\ge y} f(z) &\leq \inf_y\sup_{z\ge y} f(z)&&\qquad \text{by definition of the g.l.b.}\\ \sup_y\inf_{z\ge y} f(z) &\leq \inf_y\sup_{z\ge y} f(z)&&\qquad \text{by definition of the l.u.b.} \end{align*}