Rational Sequence Increasing to Real Number

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

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

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

where $\left\lfloor{ \cdot }\right\rfloor$ denotes the floor function.

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

From Real Number Between Floor Functions:
 * $nx - 1 < \left\lfloor{ nx }\right\rfloor \leq nx $

Thus:
 * $\dfrac{nx - 1}{n} < \dfrac{\left\lfloor{ nx }\right\rfloor}{n} \leq x $

Further:

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

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

Since $\left\langle{ x_n }\right\rangle$ is bounded above by $x$, $\left\langle{ x_{n_k} }\right\rangle$ must be increasing.

Hence the result.