Multiplying a infinite number with a rational number?

Question

Please do not down vote this question. It may be stupid, but I wonder. Why is it that we cannot multiply $3.99999\cdots$ by $4$ and write $16,....$?

Answer 1

Because it's not $12$ but $16$ ( Previously it was written as $12$ )

$$$$

Way 1

$$\begin{align}x&=3.99999....\\ 10x&=39.99999....\\ 9x&=36\\ x&=\frac{36}{9}\\ x&=4\\ 4x&=4\times4\\ 4x&=16\\ \end{align}$$

Way 2

$$\begin{align}x&=3.9999\cdots\\x=3+\frac{9}{10}+\frac{9}{100}+\cdots\\ x-3&=\frac{9}{10}+\frac{9}{100}+\cdots\\ x-3&=\frac{\frac{9}{10}}{1-\frac{1}{10}}\\ &=1\\ x&=4\\ \end{align}$$

Answer 2

Hint:

$u_n=3.\underbrace{999\ldots9}_{n- \text{times}}$

$U=\{u_n\mid n\in \mathbb{N}\}\implies\sup U=4$