Can every countable number be represented in all numeral systems?

[Wminus]

Hi. This might be a stupid question (I'm studying engineering :p), but how do you prove that all numeral systems (binary, ternary etc.) can represent every countable number?

I guess you will need to prove that any number $N$ can be written as $N= S^0 n_0 + S^1 n_1 + S^2 n_2 + ...$ where $S$ is the base of the numeral system, and $n_i \in [0,max\{S\}]$ with $i \in \mathbb{N}$.

EDIT: fixed an error in my equation

 

[Wminus]

Is the question unclear in some way? btw n_i should be element of [0,S], not [0, max{S}]. Duno why I wrote max S, I guess I'm just exhausted due to the exams.

 

[suremarc]

Hi. This might be a stupid question (I'm studying engineering :p), but how do you prove that all numeral systems (binary, ternary etc.) can represent every countable number?

I guess you will need to prove that any number $N$ can be written as $N= S^0 n_0 + S^1 n_1 + S^2 n_2 + ...$ where $S$ is the base of the numeral system, and $n_i \in [0,max\{S\}]$ with $i \in \mathbb{N}$.

EDIT: fixed an error in my equation

You can do a proof by induction--show that if some N has an expansion, then N+1 also has an expansion. Demonstrating that the process of "carrying" in addition terminates after a finite number of steps is sufficient.