Please Login to access more options.


Today

« July 2018 »

Sun

Mon

Tue

Wed

Thu

Fri

Sat

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

Problem 76: (Limit Of Quotient Equals Quotient Of Limits)

Suppose $(a_n)$ converges to $A$ and $(b_n)$ converges to $B\neq 0$, and also suppose $b_n\neq 0$ for every natural number $n$. Prove that $(a_n/b_n)$ converges to $A/B$.

Definition (Increasing Decreasing Monotonic Sequences)

Let $(a_n)$ be a sequence of real numbers.

  • We say that $(a_n)$ is (strictly) increasing if $a_n<a_{n+1}$ for every $n\in\mathbb{N}$.
  • We say that $(a_n)$ is (strictly) decreasing if $a_n>a_{n+1}$ for every $n\in\mathbb{N}$.
  • We say that $(a_n)$ is nonincreasing if $a_n\geq a_{n+1}$ for every $n\in\mathbb{N}$.
  • We say that $(a_n)$ is nondecreasing if $a_n\leq a_{n+1}$ for every $n\in\mathbb{N}$.
  • We say that $(a_n)$ is monotonic if $(a_n)$ is either nonincreasing or nondecreasing.

You should notice that a strictly decreasing sequence is nonincreasing, and a strictly increasing sequence is nondecreasing.

Problem 77: (Monotonic Sequences Converge If And Only If Bounded)

Let $(a_n)$ be a monotonic sequence. Prove that $(a_n)$ converges if and only if $(a_n)$ is bounded.

Definition (Diverges To Infinity)

Let $(a_n)$ be a sequence of real numbers. We say that $(a_n)$ diverges to infinity if for every $V$, there exists $H$ such that for every $n\in \mathbb{N}$ we know $n>H$ implies $a_n>V$. When $(a_n)$ diverges to infinity, we write $(a_n)\to \infty$.

Think of $V$ as vertical and $H$ as horizontal.

Problem 79: (Diverges To Negative Infinity)

Construct a definition of what it means to diverge to $-\infty$, by appropriately modifying the definition of diverges to $\infty$. Then prove that the sequence $(-n^3+2n)$ diverges to $-\infty$.

From Ben: The problems that remain are all designed to help you become more familiar with working with intersections and unions of arbitrarily many sets.
In the rules for set complements theorem, we prove that the complement of a union is the intersection of the complements, and we proved that the complement of an intersection was the union of the complements. These are often called De Morgan's laws. The next problem has you prove De Morgan's laws are true regardless of the number of sets involved in the union or intersection.

Problem 80: (DeMorgan's Laws For Sets)

Suppose that for each $j$ in some nonempty set $J$ that $A_j$ is a set, and also suppose that $B$ is a set. Pick one of the statements below and prove that it is true. The other is very similar.

  • $\ds B\setminus \left(\bigcup_{j\in J}A_j\right) = \bigcap_{j\in J}\left(B\setminus A_j\right) $
  • $\ds B\setminus \left(\bigcap_{j\in J}A_j\right) = \bigcup_{j\in J}\left(B\setminus A_j\right) $
One of the statements in the next problem is false. I'll let you find it.

Problem 81: (Additional Properties Of Cartesian Products)

Let $A$, $B$, $C$, and $D$ be sets. Prove or disprove each statement.

  1. $(A\times B)\cap(C\times D) = (A\cap C)\times(B\cap D)$
  2. $(A\times B)\cup(C\times D) = (A\cup C)\times(B\cup D)$
Definition (Interior And Closure)

Let $A$ be a subset of the real numbers.

  • We use the notation $A'$ to denote the set of limit points of $A$.
  • The closure of $A$, written $\text{cl}(A)$, is the union of $A$ and its limit points, so $\text{cl}(A) = A\cup A'$.
  • The interior of $A$, written $\text{int}(A)$, is the collection of interior points of $A$.

Problem 82: (The Closure of $A$ Is a Closed Set.)

Let $A$ be a set. Prove that the closure of $A$ is a closed set.

Problem 83: (The Interior Is The Union Of Every Open Set Inside A Set)

Let $S^\circ$ be the union of every open set contained in $S$. Prove that $\text{int} (S)=S^\circ$.

Problem 84: (The Closure Is The Intersection Of Every Closed Set Containing A Set)

Let $\bar A$ be the intersection of every closed set that contains $A$. Prove that $\bar A=\text{cl}(A)$.

Exercise.Standard Induction Argument

We have proven many facts this semester about how to combine two things to obtain something new. There is a very standard induction argument that allows you to take a statement, like any of the ones below, and make the statement true for any finite number of things.

  1. If $f$ and $g$ are surjective, then $f\circ g$ is surjective.
  2. If $f$ and $g$ are injective, then $f\circ g$ is injective.
  3. If $U$ and $V$ are open, then $U\cap V$ is open.
  4. If $U$ and $V$ are closed, then $U\cap V$ is closed.
  5. If $U$ and $V$ are open, then $U\cup V$ is open.
  6. If $U$ and $V$ are closed, then $U\cup V$ is closed.
  7. If $(a_n)$ converges to $A$ and $(b_n)$ converges to $B$, then $(a_n+b_n)$ converges to $A+B$.
  8. If $(a_n)$ converges to $A$ and $(b_n)$ converges to $B$, then $(a_nb_n)$ converges to $AB$.
  9. If $A_1\subseteq X$ and $A_2\subseteq X$, then we have $f(A_1\cap A_2)\subseteq f(A_1)\cap f(A_2)$.
  10. If $A_1\subseteq X$ and $A_2\subseteq X$, then we have $f(A_1\cup A_2)=f(A_1)\cup f(A_2)$.
  11. If $B_1\subseteq Y$ and $B_2\subseteq Y$, then we have $f^{-1}(B_1\cap B_2)=f^{-1}(B_1)\cap f^{-1}(B_2)$.
  12. If $B_1\subseteq Y$ and $B_2\subseteq Y$, then we have $f^{-1}(B_1\cup B_2)=f^{-1}(B_1)\cup f^{-1}(B_2)$.
  13. If $x,y\in\mathbb{R}$ then $|x+y|\leq |x|+|y|$.
  14. If $A,B,C$ are sets, then $(A\cap B)\times C = (A\times C)\cap(B\times C)$.
  15. If $A,B,C$ are sets, then $(A\cup B)\times C = (A\times C)\cup(B\times C)$.
  16. If $A,B,C$ are sets, then $A\setminus(B\cap C) = (A\setminus B)\cup(A\setminus C)$.
  17. If $A,B,C$ are sets, then $A\setminus(B\cup C) = (A\setminus B)\cap(A\setminus C)$.
  18. If $A,B,C$ are sets, then $A\cup (B\cap C)=(A\cup B)\cap(A\cup C)$.
  19. If $A,B,C$ are sets, then $A\cap (B\cup C)=(A\cap B)\cup(A\cap C)$.

For more problems, see AllProblems