Homework 3: Real numbers. Limits
Honors Introduction to Analysis 1 (MATH4130)
N.Goncharuk [email protected]
Due February 16, 2017
2.3.3.3
2.3.3.4
Task 1. Prove that the product and the sum of positive real numbers is positive.
Task 2. Consider the set of nested segments Ik = [ak , bk ] with rational endpoints (nested
means that Ik+1 ⊂ Ik for all k). Suppose that bk − ak tends to zero.
(a) Prove that ∩Ik is not empty (i.e. for some real x, for all k we have ak < x < bk ).
Hint: find a Cauchy sequence that represents x.
(b) Prove that ∩Ik contains only one real point.
(c) Is this statement correct for nested intervals (ak , bk )?
Task 3. (contains four statements, so goes for four tasks) Prove Theorem 2.3.2 for limits
of real sequences.
Task 4. Consider real numbers of the form 0.a1 a2 a3 . . . , where all ai are zeroes and ones.
Prove that there is an uncountable set of such real numbers.
Hint: we proved that the set of subsets of N is uncountable. Come up with a correspondence between subsets of N and such sequences 0.a1 a2 a3 . . . .
This task implies that R is uncountable.
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Supplementary part (please staple separately)
Task 5. Banach and Masur1 play the following game on the segment [0, 1]. Banach chooses
a segment I1 ⊂ [0, 1] with rational endpoints of length 1/2, then Masur chooses a segment
I2 ⊂ I1 with rational endpoints of length 1/4, then Banach chooses a segment I3 ⊂ I2 of
length 1/8 and so on. The game is infinite. Finally, Banach wins if the intersection ∩Ii of
all segments is the irrational point, and Masur wins if it is rational.
One of the players has a winning strategy. Find it.
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Stefan Banach (1892 – 1945) and Stanislaw Mazur (1905 – 1981) - Polish mathematicians. The task
describes a particular case of Banach-Masur game.
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