If you let be the binary expansion, then the sequence goes like
...and so forth. Choose , which is aperiodic and nonterminating, so is indeed irrational. Then for all n.
i just forget how it was worked out
Choose any irrational number x in the interval (0,1). Construct a series {X(i)} as follows: X(0)=x and Xi=(2*X(i-1))mod1 for i=1,2,3... so that the whole series is contained in (0,1). The question is the series thus constructed dense in (0,1)? why?