- #1

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I can think of an indirect proof:

1st step: make (0,1) ~ N , using a tangent function that is a 1-1 mapping from N to (0,1).

2nd step: since (0,1) is a subset of [0,1], if (0,1) is uncountable, then [0,1] must be uncountable

Problem: But these two steps doesn't necessarily mean (0,1)~[0,1], how can I resolve that? Even better, is there a direct proof? I really can't think of f such that f(0.0000...1)=0