Prove that the function defined to be 1 on the Cantor
set and 0 on the complement of the Cantor set is discontinuous at
each point of the Cantor set and continuous at every point of the
complement of the Cantor set.
Prove for the following:
a. Theorem: (Cantor-Schroder-Bernstein in the 1800s) For any set
S, |S| < |P(S)|.
b. Proposition N×N is countable.
c. Theorem: (Cantor 1873) Q is countable. (Hint: Similar. Prove
for positive rationals first. Then just a union.)
1.- Prove that the set of irrational numbers is uncountable by
using the Nested Intervals Property.
2.- Apply the definition of convergent sequence, Ratio Test or
Squeeze Theorem to prove that a given sequence converges.
3.- Use the Divergence Criterion for Sub-sequences to prove that
a given sequence does not converge.
Subject: Real Analysis
(Advanced Calculus and Real Analysis) - Cantor set,
Cantor function
* (a) Define the Cantor function.
(b) Prove that the Cantor function is
non-decreasing.
Suppose that x is real number. Prove that x+1/x =2 if and only
if x=1.
Prove that there does not exist a smallest positive real number.
Is the result still true if we replace ”real number” with
”integer”?
Suppose that x is a real number. Use either proof by
contrapositive or proof by contradiction to show that x3 + 5x = 0
implies that x = 0.