This set is due Monday, September 16, at the beginning of lecture.
Recall that . Given a sequence of nonnegative real numbers, for a finite subset of , the expression
has what is hopefully the obvious meaning: If is the increasing enumeration of the elements of , then
with the (standard) convention that if is empty, then .
For an arbitrary subset of (so may be finite or infinite), define
provided that the supremum exists. There is a small ambiguity here, in that if is finite, we have defined in two potentially conflicting ways.
1. Show that both definitions coincide if is finite.
2. Give an example of a sequence and a set such that is not defined. Show that for any and any , if is not defined, then neither is .
3. Show that, if is defined, then
More generally, show that, as long as is defined, then
and that, if this supremum exists, then so does , and the displayed equality holds.
4. Fix a positive integer . Show that if is such that, for every , has the form where then, for any , is defined, and is a number in the interval .
5. Show that for every and every positive integer there is some as in item 4. such that Describe as precisely as possible all the quadruples such that is an integer, , are sequences as in 4., and yet
Hopefully it is clear that all we are describing is the base representation of any number .
6. Indicate how to extend the above so any real has a base representation (for any ).
7. Given , let be the sequence with -th term for all . Show that is the only value of such that there are with Describe all such pairs . Show that for all there is some as in 4., with the same “failure of injectivity” property.
The above gives us that in the sense that there is an injection .
8. Make this explicit, that is, give an example of such an injection , hopefully related to these sums we are considering.
One can also show that and in fact there is a bijection between these two sets, though you do not need to do this here.
As indicated in item 7., when the function given by is not an injection.
9. For this , show that the collection of sets such that there is a set with is countable. Show that if is countable, then there is a bijection between and so, in particular, even allows us to verify that .