Let $E$ be a Riesz space. Show that if $f_n \to f$, then $f_{n_k} \to f$ for every subsequence $(f_{n_k}:k=1,2,\ldots)$ such that $n_1<n_2<\cdots$.
The reference/book was I used is "Introduction to Operator Theory in Riesz Space (1997), by Adriaan C. Zaanen" in the chapter $4$: subchapter $10$. Given below are some basic definitions.
Definition. Let $E$ be a Riesz space. The sequence $(f_n: n=1,2,\ldots)$ in $E$ is said to be increasing if $f_1 \le f_2 \le \cdots$ and decreasing if $f_1 \ge f_2 \ge \cdots$, which are denoted by $f_n \uparrow$ and $f_n \downarrow$, respectively. If $f_n \uparrow$ and $f=\sup f_n$ exists in $E$, we write $f_n \uparrow f$. Similarly, if $f_n \downarrow$ and $f=\inf f_n$ exists in $E$, we write $f_n \downarrow f$. If $f_n \uparrow f$ or $f_n \downarrow f$, we say that $f_n$ converges monotonely to $f$ as $n$ tends to infinity.
Definition. The sequence $(f_n:n=1,2,\ldots)$ in a Riesz space $E$ is said to converge in order to $f$ if there exists a sequence $p_n \downarrow 0$ such that $|f-f_n| \le p_n$ for all $n \in \Bbb N$.
Attempt: Let $f_n \to f$. By definition, there exists a sequence $(p_n)$ with $p_n \downarrow 0$ such that $|f-f_n| \le p_n$ for all $n \in \Bbb N$. So, for all $n \in \Bbb N$, we have \begin{align*} |f-f_n| \le p_n \iff f-p_n \le f_n \le f+p_n. \qquad (1) \end{align*} Now, let $(f_{n_k}:k=1,2,\ldots)$ be a subsequence of $(f_n)$ such that $n_1<n_2<\cdots$. It's clear that $f_{n_k} \in (f_n)$, for any fixed positive integer $n_k$ and for all $n \in \Bbb N$. Hence, $f_{n_k}$ must satisfy $(1)$. That is, \begin{align*} f-p_{n_k} \le f_{n_k} \le f+p_{n_k} \iff |f-f_{n_k}| \le p_{n_k}. \end{align*} Next, it's also clear that $p_{n_k} \in (p_n)$ for any fixed positive integer $n_k$ and for any $n \in \Bbb N$. Now, since $p_n \downarrow 0$, then $p_{n_k} \downarrow 0$. Hence, we found a sequence $(p_{n_k})$ with $p_{n_k} \downarrow 0$ such that $|f-f_{n_k}| \le p_{n_k}$, for all $k \in \Bbb N$. Thus, $f_{n_k} \to f$, by definition. Q.E.D.
Does the above approach correct? If no, how to approach the solution of the given problem? Any ideas? Thanks in advanced.