Not sure how to go about this one.
If pn is a non-decreasing sequence and there is a point, p, to the right of each point of the sequence, then the sequence converges to some point.
Isn't this the Monotone convergence theorem?
"If a sequence is monotone and bouned, then is converges"
here is an outline:
Suppose the sequence $\displaystyle (p_{n})$ is increasing and bounded above.
Consider the set of all points in the sequece $\displaystyle \{p_{n}|n\in\mathbb{N}\}$.
By the axiom of completeness, $\displaystyle sup\{p_{n}|n\in\mathbb{N}\}$ exists (since the set is a bounded subset of $\displaystyle \mathbb{R}$.
Assume $\displaystyle lim(p_{n})=s$, where $\displaystyle s=sup\{p_{n}|n\in\mathbb{N}\}$.
Then, for $\displaystyle \epsilon>0$, $\displaystyle s-\epsilon<s$. Thus, there exists a point $\displaystyle p_{N}$ such that $\displaystyle s-\epsilon<p_{N}<s$. Furthermore, because our sequence is increasing, it follows that, for every $\displaystyle n\geq\\N$, $\displaystyle p_{N}\leq\\p_{n}$. Thus, $\displaystyle s-\epsilon<p_{N}\leq\\p_{n}<s<s+\epsilon$.
This implies $\displaystyle |p_{n}-s|<\epsilon$