ʻAʻohe \(n \in \mathbb{Z}^+\) e like \(0 < n < 1\) .
Hōʻoia: E manaʻo kākou he wahaheʻe kēia ʻōlelo. A laila aia kahi \(n \in \mathbb{Z}^+\) e like me \(0 < n < 1\) . E noʻonoʻo kākou i ka pūʻulu \(S := \{ m \in \mathbb{Z}^+ : 0 < m < 1 \}\) . No ka mea \( n \in S\) , \(S\) 'a'ole nele. Ma muli o ke kumu hoʻonohonoho maikaʻi (ʻo kēlā me kēia ʻāpana ʻole o \(\mathbb{Z}^+\) he mea liʻiliʻi loa ia), pono e loaʻa i \(S\) kahi mea liʻiliʻi loa, ʻo ia hoʻi \(b := min(S)\) . A laila \(b \in S\) , ʻo ia ʻo \(b \in \mathbb{Z}^+\) a \(0 < b < 1\) . Hāʻawi ka huina kūpono i hoʻonui ʻia me ka huina kūpono, no laila \(b^2 \in \mathbb{Z}^+\) . Inā hoʻonui ʻoe i \(b\) me \(0<b<1\) , loaʻa iā ʻoe \(0<b^2<b\) , no laila \(0<b^2<1\) - no laila \(b^2 \in S\) . Akā, ʻoiai \(b = min(S)\) , loaʻa iā mākou \(b \leq b^2\) , e kūʻē ana i \(b^2 < b\) . No laila, ma hope o ka hōʻoia ʻana ma ke kūʻē ʻana, he ʻoiaʻiʻo ka ʻōlelo.