Unformatted text preview: ercies that appear here require the student to be familiar with the concept of
1. a. Given that S is an uncountable set of real numbers, explain why there must exist an integer n such that the
set S n, n 1 is uncountable. Solution: If the set S n, n 1 were countable for every n then, since
S Ý S n, n 1 , n Ý the set S, being the union of a countable family of countable sets, would be countable by an earlier theorem.
b. Prove that every uncountable set of real numbers must have a limit point. Hint: Apply part a and the Bolzano-Weierstrass theorem to an uncountable set of the form
2. n, n 1 . a. Suppose that S is a set of real numbers, that 0 and that for any two different members x and t of the
set S we have |x t |
. Prove that S is a countable set.
b. Suppose that S is an uncountable set of real numbers and, for each positive integer n, suppose that
1 for every y S
S n x S |x y |
Prove that each set S n is countable and that
LS S Ý Sn.
n1 c. Improve Exercise 1b by showing that if S is an uncountable set of real numbers then S has an uncountable
set of limit points that belong to S. Solution: We deduce from part b that the set S
and since the set S is countable, the set S
3. Suppose that S is a set of real numbers and that LS L S is countable. Therefore, since
ÞS LS L S must be uncountable.
0. For each integer n, in the event that the set 152 xSn
is nonempty, suppose that a number x n has been chosen in this set. Prove that for every member x of the set S
there is an integer n for which the number x n is defined and for which
, xn .
0 then there exists a countable subset E of S such that 4. Prove that if S is any set of real numbers and xn S , xn . xE 5. Suppose that S is a set of real numbers.
a. Prove that for every positive integer n it is possible to find a countable subset E n of the set S such that
x 1 ,x 1
x En b. Prove that there exists a countable subset E of S such that S E. 6. Prove that every closed subset of R is the closure of some countable set. Exercises on Upper and Lower Limits
1. Prove that if x n is a sequence of real numbers then x n has a limit if and only if
lmsup x n lminf x n .
nÝ nÝ Solution: This exercises follows at once from an earlier theorem.
2. Prove that a sequence x n of real numbers is bounded above if and only if
lmsup x n Ý.
nÝ We know that a sequence is bounded above if and only if Ý is not a partial limit of the sequence.
There a sequence is bounded above if and only if its largest partial limit is not Ý.
3. Suppose that x n is a sequence of real numbers.
a. Prove that if x is a partial limit of the sequence x n then the number x is a partial limit of the sequence
if Ý is a partial limit of a sequence x n then x n is unbounded above and so x n is
unbounded below, making Ý a partial limit of x n .
Now suppose that a real number x is a partial limit of a given sequence x n . To show that the
number x is a partial limit of the sequence x...
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