Rational Sequence Decreasing to Real Number

Theorem
Let $x \in \R$ be a real number.

Then there exists some decreasing rational sequence that converges to $x$.

Proof
Let $\left\langle{x_n}\right\rangle$ denote the sequence defined as:
 * $\forall n \in \N : x_n = \dfrac {\left\lceil{n x}\right\rceil} n$

where $\left\lceil{n x}\right\rceil$ denotes the ceiling of $n x$.

We see immediately that $\left\langle{x_n}\right\rangle$ is a rational sequence.

From Real Number between Ceiling Functions:
 * $n x < \left\lceil{n x}\right\rceil \le n x + 1$

Thus:
 * $x < \dfrac{\left\lceil{n x}\right\rceil} n \le \dfrac {n x + 1} n$

Further:

Thus, from the Squeeze Theorem for Sequences of Real Numbers:
 * $\displaystyle \lim_{n \mathop \to \infty} \frac {\left\lceil{n x}\right\rceil} n = x$

From Peak Point Lemma, there is a monotone subsequence $\left\langle{x_{n_k} }\right\rangle$ of $\left\langle{x_n}\right\rangle$.

We have that $\left\langle{x_n}\right\rangle$ is bounded below by $x$.

Hence $\left\langle{x_{n_k} }\right\rangle$ is decreasing.

Hence the result.