I'm having trouble seeing where this goes wrong.
I count by order of magnitude: $0.1,0.2,0.3 ... 0.9$. Then I go back and do $0.11,0.12,..,0.19$, and $0.21,0.22,..,0.29$, all the way to $0.91,0.92,..,0.99$ then go down an order of magnitude again $0.101, 0.102$ and so on. Like this I count all numbers on (0,1)
I suspect I miss infinite decimal expansions, but it isn't a very satisfying reason alone. For example if I argued: If I miss decimal expansions, it doesn't work because I miss some reals. If I do hit decimal expansions e.g $0.33333... = \frac{1}{3}$, then I counted $0.1999...=0.2$ twice, so our mapping isn't 1-1.
Is this reasoning sufficient?
PS I know (0,1) is uncountable, please don't mark this as a duplicate of other questions proving that it is uncountable, especially if they don't have accepted answers...