Suppose that $$\phi(n)$$ is a function of the positive integral variable $$n$$. The aggregate of all the values $$\phi(n)$$ 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 $$\phi(n)$$ is bounded above, or bounded below, or bounded. If $$\phi(n)$$ is bounded above, that is to say if there is a number $$K$$ such that $$\phi(n) \leq K$$ for all values of $$n$$, then there is a number $$M$$ such that

(i) $$\phi(n) \leq M$$ for all values of $$n$$;

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

(i) $$\phi(n) \geq m$$ for all values of $$n$$;

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

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