$a\in\mathbb{R}\setminus\mathbb{Q}$ olsun.
$$\begin{array}{rcl} (n\in\mathbb{N})(a\in\mathbb{R}\setminus\mathbb{Q}) & \Rightarrow & a\cdot 10^n < \lceil a\cdot 10^n\rceil < a\cdot 10^n+1 \\ \\ &\Rightarrow & a=\frac{a\cdot 10^n}{10^n} < \frac{\lceil a\cdot 10^n\rceil}{10^n} < \frac{a\cdot 10^n+1}{10^n}=a+\frac{1}{10^n} \end{array}$$
ve
$$\lim\limits_{n\to \infty} a=a=\lim\limits_{n\to \infty}\left(a+\frac{1}{10^n}\right)$$ olduğundan Sıkıştırma Teoremi gereğince
$$\lim\limits_{n\to \infty}\frac{\lceil a\cdot 10^n\rceil}{10^n}=a$$ olur.