ʻIke ʻia nā helu kūpono \(\mathbb{Q}\) i ka nānā mua ʻana he ʻano holoʻokoʻa mau: ma waena o nā ʻāpana ʻelua, aia mau kekahi. Akā, he hoʻopunipuni kēia manaʻo: aia nā hui o nā helu kūpono i kaupalena ʻia, akā ʻaʻole i loaʻa ko lākou supremum a i ʻole infimum ma \(\mathbb{Q}\) . Aia ke kumu i ka noho ʻana o nā helu kūpono ʻole e like me \(\sqrt{2}\) , ʻo ia hoʻi, ma kekahi ʻano, e hana ana i nā lua i ʻike ʻole ʻia ma ka laina helu kūpono.
No ka hōʻike ʻana ʻaʻole i piha ʻo \(\mathbb{Q}\) , e kuhikuhi mākou i ʻelua mau ʻāpana hakahaka ʻole o \(\mathbb{Q}\) : hoʻokahi i kaupalena ʻia ma luna akā ʻaʻohe supremum ma \(\mathbb{Q}\) , a ʻo kekahi i kaupalena ʻia ma lalo akā ʻaʻohe infimum ma \(\mathbb{Q}\) .
E noʻonoʻo i ka set \(X = \{x \in \mathbb{Q} \mid x \geq 0,\; x^2 < 2\}\) . ʻAʻole hakahaka ka set \(X\) , ʻoiai \(1 \in X\) . Inā \(x \geq 2\) , a laila \(x^2 \geq 4 > 2\) , ka mea e kūʻē ana iā \(x^2 < 2\) . No laila, paʻa ʻo \(x < 2\) no kēlā me kēia \(x \in X\) . No laila \(X \subset [0, 2]\) , a \(2\) kekahi o nā palena kiʻekiʻe o \(X\) . No laila, ua palena ʻia \(X\) ma luna.
E hoʻohana i \(x \in X\) . Hōʻike mākou aia kahi \(n \in \mathbb{N}\) i hiki ai iā \(x + \frac{1}{n} \in X\) . Ua ʻike ʻia
\[\left(x + \frac{1}{n}\right)^2 = x^2 + \frac{2x}{n} + \frac{1}{n^2} \leq x^2 + \frac{2x}{n} + \frac{1}{n} = x^2 + \frac{1}{n}(2x + 1).\]
No laila \(x^2 + \frac{1}{n}(2x + 1) < 2 \iff \frac{1}{n} < \frac{2 - x^2}{2x + 1}\) . No ka mea, ʻo \(2 - x^2 > 0\) a me \(2x + 1 > 0\) , hōʻoia ka axiom a Archimedes i ke ola ʻana o \(n \in \mathbb{N}\) . No laila, \(\left(x + \frac{1}{n}\right)^2 < 2\) , a pēlā \(x + \frac{1}{n} \in X\) .
I kēia manawa, e noʻonoʻo i ka set \(Y = \{y \in \mathbb{Q} \mid y > 0,\; y^2 > 2\}\) . ʻIke loa, \(Y \neq \varnothing\) . Eia kekahi, ua palena ʻia \(Y\) ma lalo a ua palena ʻole ʻia ma luna, ʻo ia hoʻi \(Y \subset (0, +\infty)\) . E hoʻāʻo \(y \in Y\) . Hōʻike mākou aia kahi \(m \in \mathbb{N}\) i hiki ai iā \(y - \frac{1}{m} \in Y\) . Ua ʻike ʻia
\[\left(y - \frac{1}{m}\right)^2 = y^2 - \frac{2y}{m} + \frac{1}{m^2} > y^2 - \frac{2y}{m}.\]
No laila \(y^2 - \frac{2y}{m} > 2 \iff \frac{1}{m} < \frac{y^2 - 2}{2y}\) . ʻOiai ʻo \(y^2 - 2 > 0\) a me \(2y > 0\) , hōʻoia hou ka axiom a Archimedes i ke ola ʻana o kahi \(m \in \mathbb{N}\) . No laila, \(\left(y - \frac{1}{m}\right)^2 > 2\) , a pēlā \(y - \frac{1}{m} \in Y\) .
Manaʻo ʻia ʻo \(w = \sup X\) . ʻAʻole hiki ke lilo i \(w^2 < 2\) , no ka mea, a laila \(w \in X\) , a e loaʻa kahi \(n \in \mathbb{N}\) me \(w + \frac{1}{n} \in X\) a me \(w < w + \frac{1}{n}\) , ka mea e kūʻē ana i ka ʻoiaʻiʻo ʻo \(w\) kahi palena kiʻekiʻe o \(X\) . ʻAʻole hiki ke ʻoiaʻiʻo ʻo \(w^2 > 2\) , no ka mea, a laila \(w \in Y\) , a e loaʻa kahi \(m \in \mathbb{N}\) me \(w - \frac{1}{m} \in Y\) a me \(w - \frac{1}{m} < w\) , ka mea e kūʻē ana i ka ʻoiaʻiʻo ʻo \(w\) ka palena kiʻekiʻe liʻiliʻi loa o \(X\) .
E manaʻo kākou i kēia manawa ʻo \(v = \inf Y\) . ʻAʻole hiki ke ʻoiaʻiʻo ʻo \(v^2 > 2\) , no ka mea, a laila \(v \in Y\) , a me \(v\) ʻaʻole ia he palena haʻahaʻa o \(Y\) . Pēlā nō, ʻaʻole hiki ke ʻoiaʻiʻo ʻo \(v^2 < 2\) , no ka mea, a laila \(v \in X\) , a pēlā ʻaʻole ʻo \(v\) ka palena haʻahaʻa nui loa o \(Y\) .
No laila, ʻo nā mea hiki wale nō i koe ʻo \(w^2 = 2\) a me \(v^2 = 2\) . Eia nō naʻe, ʻaʻohe helu rational nona ka huinahā like me \(2\) . No laila \(w = \sup X\) ke noho ma \(\mathbb{Q}\) . No ke kumu hoʻokahi, ʻaʻole hiki iā \(v = \inf Y\) ke noho ma \(\mathbb{Q}\) .
No laila, ʻaʻole \(\mathbb{Q}\) he kahua i hoʻonohonoho pono ʻia. \(\square\)