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 \[k \leq m \leq M \leq K.\]

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