Question

**Problem 1.** Let
{*E*_{n}}_{n}^{∞}_{=1} be a
sequence of nonempty (Lebesgue) measurable subsets of [0, 1]
satisfying

lim_{n→∞}*m*(*E*_{n}) = 1.

Show that for each ε ∈ [0, 1) there exists a subsequence
{*E*_{nk} }_{k}^{∞}_{=1} of
{*E*_{n}}_{n}^{∞}_{=1} such
that
*m*(∩_{k}^{∞}_{=1}*E*_{nk})
≥ ε

Answer #1

Question 4 Let a sequence {an}∞ n=1 a sequence satisfying the
condition ∀c ∈ (0, 1), ∀n ∈ N, | a_(n+2) − a_(n+1) | < c
|a_(n+1) − a_n |.
4.1 Show that
∀c ∈ (0, 1), ∀n ∈ N, n ≥ 2, | a_(n+1) − a_n | < c^(n−1) | a_2
− a_1 |.
4.2 Show that {an}∞ n=1 is a Cauchy sequence

Discrete mathematics function relation
problem
Let P ∗ (N) be the set of all nonempty subsets of N. Define m :
P ∗ (N) → N by m(A) = the smallest member of A. So for example, m
{3, 5, 10} = 3 and m {n | n is prime } = 2.
(a) Prove that m is not one-to-one.
(b) Prove that m is onto.

Let X be a set and let (An)n∈N be a sequence of subsets of X.
Show that: (a) If (An)n∈N is increasing, then liminf An = limsupAn
=S∞ n=1 An. (b) If (An)n∈N is decreasing, then liminf An = limsupAn
=T∞ n=1 An.

Problem 1 Let {an} be a decreasing and bounded
sequence. Prove that limn→∞ an exists and
equals inf{an}.

Telescoping Series. Let {an} ∞ n=0 be a sequence of real numbers
converging to zero, limn→∞ an = 0. Let bn = an − an+1. Then the
series X∞ n=0 bn converges.

Definition: Let p be a prime and 0 < n then the p-exponent of
n, denoted ε(n, p) is the largest number k such that pk | n.
Note: for p does not divide n we have ε(n,p) = 0
Notation: Let n ∈ N+ we denote the set {p : p is prime and p |
n} by Pr(n). Observe that Pr(n) ⊆ {2, 3, . . . n} so that Pr(n) is
finite.
Problem: Let a, b be...

Define a sequence (xn)n≥1 recursively by x1 = 1 and
xn = 1 + 1 /(xn−1) for n > 0. Prove that limn→∞ xn = x exists
and find its value.

If (xn) ∞ to n=1 is a convergent sequence with limn→∞ xn = 0
prove that
lim n→∞ (x1 + x2 + · · · + xn)/ n = 0 .

Let A be an n×n matrix. If there exists k > n such that A^k
=0,then
(a) prove that In − A is nonsingular, where In is the n × n
identity matrix;
(b) show that there exists r ≤ n such that A^r= 0.

Exercise 2.4.5: Suppose that a Cauchy sequence {xn} is such that
for every M ∈ N, there exists a k ≥ M and an n ≥ M such that xk
< 0 and xn > 0. Using simply the definition of a Cauchy
sequence and of a convergent sequence, show that the sequence
converges to 0.

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