Equivalence class on real numbers Call two real numbers equivalent if their binary decimal expansion differ in a finite amount of places, if S is a set which contains an element of every equivalence class, must S contain an interval?
How to show that every interval contains an (uncountable number of?) element of every equivalence class? 

First, a technical point: for terminating expansions in binary, such as 1 or &half;, we must choose one of the two valid binary expansions for representing them: for instance, either 1.000... or 0.111... in the case of 1. This is necessary for the equivalence classes to be well-defined. We may establish the convention of always choosing the terminating expansion.
Then, you can solve your problem by making the following observations:


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*For a given equivalence class E and a fixed representative e ∈ E, how can you characterize an arbitrary x ∈ E relative to e? What are the implications for the cardinality of E?

*Is there any particularly simple sort of interval which is guaranteed to contain at least one representative from each equivalence class? Can you show that any interval strictly contains such an interval (and therefore, also contains infinitely many)?

*Consider a partitioning of ℝ into the rational and irrational numbers. Can you find a way of selecting representatives for each equivalence class of the rationals, and also for the irrationals, such that their union does not contain any intervals?
I think that this approach should be fairly straightforward.
