Suppose $f\_n$ is a sequence of real valued functions on $[0,1]$ which converges pointwise to zero. 


1. Is there an uncountable subset $A$ of $[0,1]$ so that $f\_n$ converges uniformly on $A$?
2. Is there a subset $A$ of $[0,1]$ of cardinality the continuum so that $f\_n$ converges uniformly on $A$?


Background: Egoroff's theorem implies that the answer to (2) is yes if all $f\_n$ are Lebesgue measurable. It is not hard to show that the answer to (1) is yes if you change "uncountable" to "infinite".


Motivation: I thought about this question while teaching real analysis this term but could not solve it even after looking at some books, googling, and asking some colleagues who are much smarter than I, so I assigned it as a problem (well, an extra credit problem) to my class. Unfortunately, no one gave me a solution. 


ADDED 11-12-10: Thanks for all the great answers. I accepted Jonas' answer since it was the first one.



Does pointwise convergence imply uniform convergence on a large subset?
I did some Googling and came up with something that looks relevant, [Theorem 10](https://books.google.com/books?id=WwmvxtDlz9UC&lpg=PA124&ots=lcdSy9gacd&dq=point%2520set%2520theorem%2520morgan&pg=PA88#v=onepage&q&f=false) quoted below from Morgan's *Point set theory*. It cites works of Sierpiński from the late 1930s, but I can't tell what works are cited because the preview won't let me see that page in the references.



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> The existence of a linear set having the power of the continuum that is concentrated on a denumerable set is equivalent to the existence of a pointwise convergent sequence of functions of a real variable that does not converge uniformly on any uncountable set.
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