MHB Real Analysis, Sequences in relation to Geometric Series and their sums

[joypav]

I will state the problem below. I don't quite understand what I am needing to show. Could someone point me in the right direction? I would greatly appreciate it.

Problem:

Let p be a natural number greater than 1, and x a real number, 0<x<1. Show that there is a sequence $(a_n)$ of integers with $0 \leq a_n < p$ for each n such that
$ x = \sum\limits_{n=1}^{\infty} \frac{a_n}{p^n} $

and that this sequence is unique except when x is of the form $ \frac{q}{p^n} $, in which case there are exactly two such sequences.

Show that, conversely, if $(a_n)$ is any sequence of integers with $ 0 \leq a_n < p$, the series

$ \sum\limits_{n=1}^{\infty} \frac{a_n}{p^n} $

converges to a real number x with $ 0 \leq x \leq 1 $. If p=10, this sequence is called the decimal expansion of x. For p=2 it is called the binary expansion of x, and for p=3, the ternary expansion.

 

[Euge]

For a start, let $a_1$ be the greatest integer such that $\dfrac{a_1}{p} \le x$. Having chosen $a_1,\ldots, a_{N-1}$, let $a_N$ be the greatest integer such that $\displaystyle \sum_{n = 1}^N \frac{a_n}{p^n} \le x$. Show that the sequence of partial sums $\displaystyle \left(\sum_{n = 1}^N \frac{a_n}{p^n}\right)_{N=1}^\infty$ converges to $x$.

 

[joypav]

For a start, let $a_1$ be the greatest integer such that $\dfrac{a_1}{p} \le x$. Having chosen $a_1,\ldots, a_{N-1}$, let $a_N$ be the greatest integer such that $\displaystyle \sum_{n = 1}^N \frac{a_n}{p^n} \le x$. Show that the sequence of partial sums $\displaystyle \left(\sum_{n = 1}^N \frac{a_n}{p^n}\right)_{N=1}^\infty$ converges to $x$.

Then we have,

$ 0 < x - \sum\limits_{n=1}^{N} \frac{a_n}{p^n} < \frac{1}{p^N} $

Take the $ lim_{n\rightarrow{\infty}} $,
$ \vert x - \sum\limits_{n=1}^{\infty} \frac{a_n}{p^n} \vert = 0 $

$ \implies \sum\limits_{n=1}^{\infty} \frac{a_n}{p^n} $ converges to x.

Is this correct? Now to show uniqueness do I proceed by way of contradiction?

 

[joypav]

Also, I don't understand the second part of the problem either. Now I am to find the x it converges to?