Rational Sequence Decreasing to Real Number

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Theorem

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


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


Proof

Let $\sequence {x_n}$ denote the sequence defined as:

$\forall n \in \N : x_n = \dfrac {\ceiling {n x} } n$

where $\ceiling {n x}$ denotes the ceiling of $n x$.


From Ceiling Function is Integer, $\ceiling {n x}$ is an integer.

Hence by definition of rational number, $\sequence {x_n}$ is a rational sequence.


From Real Number is between Ceiling Functions:

$n x < \ceiling {n x} \le n x + 1$

Thus:

$x < \dfrac {\ceiling {n x} } n \le \dfrac {n x + 1} n$


Further:

\(\displaystyle \lim_{n \mathop \to \infty} \frac {nx + 1} n\) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \paren {x + \frac 1 n}\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} x + \lim_{n \mathop \to \infty} \frac 1 n\) Combined Sum Rule for Real Sequences
\(\displaystyle \) \(=\) \(\displaystyle x + 0\) Quotient Rule for Limits of Functions
\(\displaystyle \) \(=\) \(\displaystyle x\)

Thus, from the Squeeze Theorem for Real Sequences:

$\displaystyle \lim_{n \mathop \to \infty} \frac {\ceiling {n x} } n = x$


From Peak Point Lemma, there is a monotone subsequence $\sequence {x_{n_k} }$ of $\sequence {x_n}$.

We have that $\sequence {x_n}$ is bounded below by $x$.

Hence $\sequence {x_{n_k} }$ is decreasing.

From Limit of Subsequence equals Limit of Real Sequence, $\sequence {x_{n_k} }$ converges to $x$.

Hence the result.

$\blacksquare$