By Chowdhury K.C.
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This is not yet a complete proof. We will now fill in the gaps. First let x E [0, 1). We introduce the functions 1 k-I I - Xk , = lim Gr(x). G(x) _ fl k-I l - Xk m-**o 1 The product defining G converges for x E [0, 1) because the series 2:Xk do. For fixed x in (0, 1), the series Gm(x) grows monotonically. Therefore, Gm(x) < G(x) for fixed x e [0, 1) and every m. Since G(x) is a product of a finite number of absolutely convergent series, Gm(x) is absolutely convergent and can be written as Gm(X) - pm(k)Xk.
We introduce the functions 1 k-I I - Xk , = lim Gr(x). G(x) _ fl k-I l - Xk m-**o 1 The product defining G converges for x E [0, 1) because the series 2:Xk do. For fixed x in (0, 1), the series Gm(x) grows monotonically. Therefore, Gm(x) < G(x) for fixed x e [0, 1) and every m. Since G(x) is a product of a finite number of absolutely convergent series, Gm(x) is absolutely convergent and can be written as Gm(X) - pm(k)Xk. k-0 where pm(k) denotes the number of partitions of k into parts not greater than m (pm(0):= 1).
Tn is called the nth convergent to the sequence ao. a,, a2, ... 15) Theorem. Let ao E Z: a, , a2, . Then the sequence . E N. n _ , 2.... converges to 0, where 0 is an irrational number. The a; are uniquely defined by the expansion of 0 as a continued fraction. Conversely, let a,,> is 0 be an arbitrary irrational number. Then 0 - lim Tn if Tn -