"Show that for every infinite language L, there exists a sub-language L'

of L that is not Turing-recognizable (specifically, L' is undecidable)"

I'm not sure if I fully understood the proof…what do you mean by "We found a correspondence between L' and B (the set of all infinite

binary sequences), which is an uncountable set."

Is L' the set of all sub-languages of L or is it a sub-language of L?

and how can you deduce from this correspondence that the set of sub-languages of L is uncountable? Did you assume that it was an onto correspondence?

Thanks in advance