Let $\left\{a_i\right\},\left\{b_i\right\}$ be two Cauchy sequences with respect to norm $|\cdot |_p$ on $\mathbb{Q},$ that is, $a_i,b_i\in \mathbb{Q},$ and $$\forall \ \varepsilon>0,\exists N\in \mathbb{N},s.t. n,m> N\Longrightarrow |a_m-a_n|_p<\varepsilon,\ |b_m-b_n|_p<\varepsilon.$$(p is a prime number.) Such two sequences $\left\{a_i\right\},\left\{b_i\right\}$ are equivalent iff $|a_i-b_i|_p\rightarrow 0.$ $\mathbb{Q}_p$ is the set of equivalence classes of such Cauchy sequences. For any equivalence class $a\in\mathbb{Q}_p,$ if $\left\{a_i\right\}\in a,$ then the norm of $a$ is defined as$$|a|_p:=\lim_{i \rightarrow \infty} |a_i|_p.$$
Question: Let $\beta_n (n\in \mathbb{N})$ be a Cauchy sequence in $\mathbb{Q}_p,$ with respect to norm $|\cdot |_p$ on $\mathbb{Q}_p.$ My textbook says that if $\left\{a_{n_i}\right\}_{i\in \mathbb{N}}\in \beta_n,$ then the equivalence class of the sequence $\left\{a_{i_i}\right\}_{i\in \mathbb{N}}$ is the limit of $\beta_n,$ which means, if $\left\{a_{i_i}\right\}_{i\in \mathbb{N}}\in c,$ then $\displaystyle\lim_{i \rightarrow \infty} |\beta_i-c|_p=0.$ But I cannot think of a proof. Any help would be appreciated.