首先一个错误得更正一下，更严谨地讲并集符号上的那个符号不是infinite的意思，而是countable的意思。
更严谨的证明如下：
For every a that belongs to (0,1),
we get 0<a<1,
according to archimedean property, we can find positive N1 and N2
s.t. 1<a*N1 and 1<(1-a)*N2. It means that 1/N1<a and a<1-1/N2.
Name N=max[N1, N2], then 1/N<=1/N1<a<1-1/N2<=1-1/N,
it means that a belongs to U[1/n, 1-1/n]

For every b that belongs to  U[1/n, 1-1/n]
that is a certain N s.t. 1/N<=b<=1-1/N
as 0<1/N<=b<=1-1/N<1
b belongs to (0,1)
Conclude all above, we know every element that belongs to (0,1) exists in  U[1/n, 1-1/n] and vice versa. so (0,1)=U[1/n, 1-1/n]