Can we find or construct a decreasing subsequence in $(x_{n})_{n≥1}$ converging to $0$ Let $(x_{n})_{n≥1}$ be a real sequence converging to $0$. My question: Can we find or construct a decreasing or increasing subsequence in $(x_{n})_{n≥1}$ converging to $0$.

The answer is yes, we will construct a decreasing or increasing sequence out of any sequence.

We call $x_n$ a point of no return if for every $m\geq n$, $x_n\geq x_m$, if $\\{x_n\\}$ has infinitely many points of no return, then we just construct the subsequence made just by the points of no return and we will have an decreasing subsequence.

If $\\{x_n\\}$ has only finitely many points of no return, then, let $x_r$ be the last point of no return, then $x_{r+1}\geq x_r$, but since it isn't a point of no return, there must be an $r_0$ such that $x_{r_0}\geq x_{r+1}$, but since $x_{r_0}$ isn't a point of no return there must be an $r_1$ such that $x_{r_1}\geq x_{r_0}$ and so on. This way we constructed an increasing subsequence.

This plus the fact that every subsequence of a convergent sequence converges to the same value, gives us what we wanted.