I stumbled upon a proof that 1 = 0.999... It's something that I have taken for granted, but the proof actually held me dazed for a short period of time. Until I remembered that I have always taken this for granted. Here we go
$$x = 0.999 \Rightarrow \\ 10x = 9.999 \Rightarrow \\ 10x = 9 + 0.999 \Rightarrow \\ 10x = 9 + x \Rightarrow \\ 9x = 9 \Rightarrow \\ x = 1 \Rightarrow \\ 0.999 = 1$$ I actually like the proof - it seems that something is wrong about our way of reasoning. But then your intuition kicks in.
$$ 0.333 = \frac{1}{3} \Rightarrow \\ 0.333 \cdot 3 = 3 \cdot \frac{1}{3} \Rightarrow \\ 0.999 = 1 $$ This proof isn't so much to my liking since the premise seems a shortcut.
Since 0.999 is a infinite series of numbers, since an infinite series of numbers can be written as an infinite sum and since this particular sum converges towards 1, we can establish the equality $1 = 0.999$. To be more exact we can obtain the sum $0.999... = 0.9 + 0.09 + 0.009 + ...$ and $$ \sum_{k = 1}^{\infty} \frac{9}{10^{k}} = lim_{n \rightarrow \infty} \left (1 - \frac{1}{10^{n}}\right ) $$ where $$ lim_{n \rightarrow \infty} \frac{1}{10^{n}} = 0 $$ As analytical proofs often goes, this is actually a bit philosophical. Liked!