MathItems

Let $\varepsilon > 0$ be given. There exists integers $N$, $N'$ such that

• $n \geq N$ implies $d(p_n,p) < \varepsilon / 2$,
• $n \geq N'$ implies $d(p_n,p') < \varepsilon / 2$.

Hence if $n \geq \max(N, N')$, we have

$$d(p,p') \leq d(p,p_n) + d(p_n,p') < \varepsilon \; .$$

Since $\varepsilon$ was arbitrary, we conclude that $d(p,p') = 0$.