I'm trying to prove the sequence ($ x_j = \sum_0^{j-1} p^k$) for $|p|<1$ is a Cauchy Sequence.
I understand that if i take $x_m$ to any element less than $1/N$, then all $j,k\geq m$ $|x_j-x_k| < 1/N$. However, i am having trouble formalizing this. Does anyone have any tips how i can go forward with this proof?
You can prove your sequence is a Cauchy sequence in two ways:
Using the fact that every convergent sequence is a Cauchy sequence. So you have to prove that $\lim\limits_{n\to +\infty}x_n$ is a real number. Notice that $\forall n\in\mathbb{N},\,x_n=\sum\limits_{k=0}^{n-1}p^k=\frac{1-p^n}{1-p}$.
Using the definition of a Cauchy sequence. Then you have to prove that:$$\forall\varepsilon>0,\,\exists N\in\mathbb{N},\,\forall m,n\ge N,\,|x_m-x_n|<\varepsilon$$ where, of course, $m,n\in\mathbb{N}$. An equivalent definition is:$$\forall\varepsilon>0,\,\exists N\in\mathbb{N},\,\forall n\ge N,\,\forall r\in\mathbb{N},\,|x_{n+r}-x_n|<\varepsilon$$Notice that if $n,r\in\mathbb{N}$, we have:$$x_{n+r}-x_n=\dfrac{1-p^{n+r}}{1-p}-\dfrac{1-p^n}{1-p}=\dfrac{1-p^r}{1-p}p^n$$
In both cases, don't forget that $|p|<1$.