Let $X$ be a metric space.
(a) Call two Cauchy sequences $\left\{ p_n \right\}$, $\left\{ q_n \right\}$ in $X$ equivalent if $$ \lim_{n \to \infty} d \left( p_n, q_n \right) = 0.$$ Prove that this is an equivalence relation.
Can someone let me know if my proof for transitivity is correct?
Let $\{p_n\}, \{q_n\}, \{r_n\}$ be Cauchy sequences in $X$. Suppose $\lim\limits_{n \to \infty} d \left( p_n, q_n \right) = 0 \textrm{ and } \lim\limits_{n \to \infty} d \left( q_n, r_n \right) = 0$. Let $\epsilon > 0$. Then, $\exists N_1, N_2 \in \mathbb{N}$ such that \begin{equation*} \begin{split} n \geq N_1 &\implies d(p_n, q_n) < \epsilon/2 \\ n \geq N_2 &\implies d(q_n, r_n) < \epsilon/2 \end{split} \end{equation*} Pick $N = \max\{N_1, N_2\}$. Then, by the triangle inequality, $n \geq N$ implies \begin{equation*} d(p_n, r_n) \leq d(p_n, q_n) + d(q_n, r_n) < \epsilon \end{equation*} showing that $p_n \to r_n$ which means that$\lim\limits_{n \to \infty} d \left(p_n, r_n \right) = 0$.
Almost. You claimed to have proved that $p_n\to r_n$, whatever that means. What you did prove was that $\lim_{n\to\infty}d(p_n,r_n)=0$, and that is what you were supposed to prove.