Re: Prove that an irrational number has an infinite base q expansion

[José Carlos Santos]

Alex wrote:

> Can anyone give me a hint as to how I would prove that an irrational
> number has an infinite base q expansion for all q>=2 ?
>
> I am sure it must be shockingly obvious but I need some inspiration!

Saying that a number _x_ has a finite expansion on base _q_ is saying
that _x_ can be written as

   a_n*q^n + a_{n - 1}q^{n - 1} + ... + a_m*q^m,

where _m_ and _n_ are integers such that m < n and each a_k is in
{0,1,...,q - 1}. But then _x_ is a sum of rational numbers and therefore
a rational number.

Best regards,

Jose Carlos Santos