Let $a$ be a set accumulation point $X$. Show that there is either an increasing sequence, or a decreasing sequence of points $x_n \in X$ with $lim_{x_n}=a$.
Let $A_n = (a -\frac{1}{n}, a)$ and $B_n = (a, a +\frac{1}{n})$, as $a \in A'$ then one of these sets has infinite elements of $A$, if $A_n$ is infinite we can define $(x_n)$ increasing with $lim x_n = a$ otherwise we define $(y_n)$ decreasing, both with limit $a$.
That's right? is enough?
Thanks.
Edit: Seems like I misread your approach a bit, you seem to want to keep picking elements $x_{n}$ from $(a_{n}-1/n, a)\cap X$(Assuming WLOG it’s always non empty), but that doesn’t necessarily give you an increasing sequence! (Thanks to BS Thomson, pointing for out I misread your argument!) What you can do to fix this is to keep picking increasing elements and now this is fixed. Also note the use of the axiom of (countable) choice.