81. The bounds of a bounded function

Suppose that $\varphi (n)$ is a function of the positive integral variable $n$. The aggregate of all the values $\varphi (n)$ defines a set $S$, to which we may apply all the arguments of § 80. If $S$ is bounded above, or bounded below, or bounded, we say that $\varphi (n)$ is bounded above, or bounded below, or bounded. If $\varphi (n)$ is bounded above, that is to say if there is a number $K$ such that $\varphi (n)\le K$ for all values of $n$, then there is a number $M$ such that

(i) $\varphi (n)\le M$ for all values of $n$;

(ii) if $ϵ$ is any positive number then $\varphi (n)>M–ϵ$ for at least one value of $n$. This number $M$ we call the upper bound of $\varphi (n)$. Similarly, if $\varphi (n)$ is bounded below, that is to say if there is a number $k$ such that $\varphi (n)\le k$ for all values of $n$, then there is a number $m$ such that

(i) $\varphi (n)\ge m$ for all values of $n$;

(ii) if $ϵ$ is any positive number then $\varphi (n)<m+ϵ$ for at least one value of $n$. This number $m$ we call the lower bound of $\varphi (n)$.

If $K$ exists, $M\le K$; if $k$ exists, $m\ge k$; and if both $k$ and $K$ exist then $$k\le m\le M\le K.$$