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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.

2 what does convergeness mean

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# Does pointwise convergeness imply uniform convergence on a large subset?

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.