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Chapter 1 Limit Points and Sequences

You will find undefined words in these notes such as collection, element, in, member and number. We assume that you have an intuitive understanding of these words and an intuitive understanding of the algebra associated with the real numbers.

Definition 1.1

By a point is meant an element of the real numbers, \(\mathbb{R}\).

Definition 1.2

By a point set is meant a collection of one or more points.

Definition 1.3

The statement that the point set \(M\) is linearly ordered means that there is a meaning for the words “less than” and “greater than” so that if each of \(a\text{,}\) \(b\text{,}\) and \(c\) is in \(M\text{,}\) then

  1. if \(a \lt b\) and \(b \lt c\) then \(a \lt c\) and

  2. one and only one of the following is true:

    1. \(a \lt b\text{,}\)

    2. \(b \lt a\text{,}\) or

    3. \(a=b\text{.}\)

When we refer to “two points,” we adhere to standard usage of the English language and thus imply that they are not the same point. For example, if you went to “two stores” we would not assume that you visited the same store twice. On the other hand if we were to write, “let each of \(a\) and \(b\) be a point” then it would be possible that \(a\) is the same point as \(b\text{.}\)

Definition 1.10

The statement that the point set \(O\) is an open interval means that there are two points \(a\) and \(b\) such that \(O\) is the set consisting of all points between \(a\) and \(b\text{.}\)

Definition 1.11

The statement that the point set \(I\) is a closed interval means that there are two points \(a\) and \(b\) such that \(I\) is the set consisting of the points \(a\) and \(b\) and all points between \(a\) and \(b\text{.}\)

In set notation,

\begin{equation*} (a,b) = \{ x: x \mbox{ is a point and } a \lt x \lt b \} \end{equation*}

and

\begin{equation*} [a,b] = \{ x: x \mbox{ is a point and } a \leq x \leq b \}. \end{equation*}

We do not use \((a,b)\) or \([a,b]\) in the case \(a = b,\) although many mathematicians and texts do. We refer to \(a\) and \(b\) as the endpoints of the interval.

Definition 1.12

If \(M\) is a point set and \(p\) is a point, the statement that \(p\) is a limit point of the point set \(M\) means that every open interval containing \(p\) contains a point of \(M\) different from \(p\text{.}\)

Problem 1.13

Show that if \(M\) is the open interval \((a,b)\text{,}\) and \(p\) is in \(M\text{,}\) then \(p\) is a limit point of \(M\text{.}\)

Problem 1.14

Show that if \(M\) is the closed interval \([a,b]\text{,}\) and \(p\) is not in \(M\text{,}\) then \(p\) is not a limit point of \(M\text{.}\)

Problem 1.15

Show that if \(M\) is a point set having a limit point, then \(M\) contains 2 points. Must \(M\) contain 3 points? 4 points?

Problem 1.16

Show that if \(M\) is the set of all positive integers, then no point is a limit point of \(M\text{.}\)

Problem 1.17

Assume \(M\) is a point set and \(p\) is a point of \(M\text{.}\) Create a definition for “\(q\) is the first point to the left of \(p\) in \(M\)” by completing the following. “If \(M\) is a point set and \(p\) is a point in \(M\text{,}\) then...”

Problem 1.18

Assume \(M\) is a point set such that if \(p\) is a point of \(M\text{,}\) there is a first point to the left of \(p\) in \(M\) and a first point to the right of \(p\) in \(M\text{.}\) Is it true that \(M\) cannot have a limit point?

Definition 1.19

If each of \(A\) and \(B\) is a set, then the set defined by \(A\) union \(B\) is the set consisting of all elements that are in \(A\) or in \(B\text{.}\)

Definition 1.20

If each of \(A\) and \(B\) are sets, then the set defined by \(A\) intersect \(B\) is the set consisting of all elements that are in \(A\) and in \(B\text{.}\)

If \(M\) is a set and \(m\) is a point then the notation \(m \in M\) translates as “m is in \(M\text{.}\)” \(A \cup B\) is written in set notation as \(A \cup B = \{ x | x \in A \mbox{ or } x \in B \}\) and \(A \cap B\) is written in set notation as \(A \cap B = \{ x | x \in A \mbox{ and } x \in B \}\text{.}\)

Problem 1.21

Show that if \(H\) is a point set and \(K\) is a point set and there is a point that is in both \(H\) and \(K\) and \(p\) is a limit point of \(H \cap K\text{,}\) then \(p\) is a limit point of \(H\) and \(p\) is a limit point of \(K\text{.}\)

Problem 1.22

Show that if \(H\) is a point set and \(K\) is a point set and every point of \(H\) is a limit point of \(K\) and \(p\) is a limit point of \(H\text{,}\) then \(p\) is a limit point of \(K\text{.}\)

Problem 1.23

If \(H\) is a point set and \(K\) is a point set and \(p\) is a limit point of \(H \cup K\text{,}\) then \(p\) is a limit point of \(H\) or \(p\) is a limit point of \(K\text{.}\)

Problem 1.24

Show that if \(M\) is the set of all reciprocals of positive integers, then \(0\) (zero) is a limit point of \(M\text{.}\)

Up until now, the word point has meant a real number. From here forward, it may also be used to mean an ordered pair of real numbers, i.e. a point in the plane.

Definition 1.25

The statement that \(f\) is a function means that \(f\) is a collection of points in the plane, no two of which have the same first coordinates.

Definition 1.26

If \(f\) is a function, then by the domain of \(f\) is meant the point set of all first coordinates of the ordered pairs in \(f\text{,}\) and by the range of \(f\) is meant the set of all second coordinates of the ordered pairs in \(f\text{.}\)

We use the usual notation that if \(f\) if a function and \(x\) is a number in the domain of \(f\text{,}\) then \(f(x)\) is the number which is the \(2^{nd}\) coordinate of the point of \(f\) whose \(1^{st}\) coordinate is \(x\text{.}\)

Definition 1.27

A sequence is a function with domain the natural numbers and with range a subset of real numbers.

If \(p\) is a sequence, then \(p = \{(1,p(1)), (2,p(2)), (3,p(3)), \ldots\}\text{.}\) Since writing \(p\) this way is cumbersome and the domain is always the natural numbers, we will denote sequences by listing only the points in the range of the sequence, \(p(1), p(2), p(3), \dots\text{.}\) We'll further abbreviate this as: \(p_1, p_2, p_3, \dots\text{.}\) The set \(\{p_i : i =1,2,3,\dots \}\) denotes the range of the sequence. That is, \(\{p_i : i =1,2,3,\dots \}\) denotes the point set to which the point \(x\) belongs if and only if there is a positive integer \(n\) such that \(x=p_n\text{.}\)

Definition 1.28

The statement that the point sequence \(p_1,p_2,\dots\) converges to the point \(x\) means that if \(S\) is an open interval containing \(x\) then there is a positive integer \(N\) such that if \(n\) is a positive integer and \(n \ge N\) then \(p_n \in S\text{.}\)

Definition 1.29

The statement that the sequence \(p_1, p_2, p_3, \ldots\) converges means that there is a point \(x\) such that \(p_1, p_2, p_3, \ldots\) converges to \(x\text{.}\)

Problem 1.30

For each positive integer \(n\text{,}\) let \(p_n= 1 - 1/n\text{.}\) Show that the sequence \(p_1,p_2,p_3,\dots\) converges to \(1\text{.}\)

Problem 1.31

For each positive integer \(n\text{,}\) let \(p_{2n-1}=1/(2n-1)\) and let \(p_{2n}=1+1/2n\text{.}\) Does the sequence \(p_1,p_2,p_3,\dots\) converge to \(0\text{?}\)

Problem 1.32

For each positive integer \(n\text{,}\) let \(p_{2n} = 1/(2n-1)\text{,}\) and let \(p_{2n-1}= 1/2n\text{.}\) Show that the sequence \(p_1,p_2,p_3,\dots\) converges to \(0\text{.}\)

Problem 1.33

Show that if the sequence \(p_1,p_2,p_3,\dots\) converges to the point \(x\text{,}\) and, for each positive integer \(n\text{,}\) \(p_n \ne p_{n+1}\text{,}\) then \(x\) is a limit point of the set which is the range of the sequence.

Problem 1.34

Show that if \(p \ne 0\text{,}\) then \(p\) is not a limit point of the set \(\{ 1 , {1 \over 2} , {1 \over 3} , \dots \}\text{.}\)

Problem 1.35

Show that if \(c\) is a number and \(p_1,p_2,p_3,\dots\) is a sequence which converges to the point \(x\text{,}\) then the sequence \(c \cdot p_1,c \cdot p_2,c \cdot p_3,\dots\) converges to \(c \cdot x\text{.}\)

Problem 1.36

Show that if the sequence \(p_1,p_2,p_3,\dots\) converges to \(x\) and the sequence \(q_1,q_2,q_3,\dots\) converges to \(y\text{,}\) then the sequence \(p_1+q_1,p_2+q_2,p_3+q_3,\dots\) converges to \(x+y\text{.}\)

Definition 1.37

The statement that \(p\) is the first point to the right of the point set \(M\) means that \(p\) is greater than every point of \(M\) and if \(q\) is a point less than \(p\text{,}\) then \(q\) is not greater than every point of \(M\text{.}\)

Definition 1.38

The statement that \(p\) is the right-most point of \(M\) means that \(p\) is in \(M\) and no point of \(M\) is greater than \(p\text{.}\)

Problem 1.39

Show that if \(M\) is a point set, then there cannot be both a right-most point of \(M\) and a first point to the right of \(M\text{.}\)

Problem 1.40

Show that if \(M\) is a point set and there is a point \(p\) which is the first point to the right of \(M\text{,}\) then \(p\) is a limit point of \(M\text{.}\)

Definition 1.42

The statement that the point set \(M\) is finite means that there is a positive integer \(n\) such that \(M\) contains \(n\) points but \(M\) does not contain \(n+1\) points.

Definition 1.43

The statement that the point set \(M\) is infinite means that \(M\) is not finite.

Definition 1.47

If \(A\) and \(B\) are point sets, then we say that \(A\) is a subset of \(B\) if every point of \(A\) is also a point of \(B\text{.}\) This is typically denoted by \(A \subseteq B\text{.}\)

Definition 1.48

The statement that the point set \(M\) is an open point set means that for every point \(p\) of \(M\) there is an open interval which contains \(p\) and is a subset of \(M\text{.}\)

Definition 1.49

The statement that the point set \(M\) is a closed point set means that if \(p\) is a limit point of \(M\text{,}\) then \(p\) is in \(M\text{.}\)

Note that if a set \(M\) has no limit point, then it is a closed point set. We could equivalently define closed by saying that \(M\) is closed if, and only if, there is no limit point of \(M\) that is not in \(M\text{.}\)

Definition 1.53

The statement that the point set \(M\) is bounded means that \(M\) is a subset of some closed interval.

Definition 1.54

Let \(M\) be a point set. The statement that \(M\) is bounded below means that there is a point \(z\) such that \(z\) is less than or equal to \(m\) for every \(m\) in \(M\text{.}\) Bounded above is defined similarly.

The following axiom will be used in many subsequent Problems. Such problems are marked with (CA). When you see (CA) you know that you must either use the Completeness Axiom or at least a problem that we have solved that uses the Completeness Axiom.

Similarly, if there is a point to the left of every point of \(M\text{,}\) then there is either a left-most point of \(M\) or a first point to the left of \(M\text{.}\)

Definition 1.58

The statement that the sequence \(p_1, p_2,p_3,\dots\) is an increasing sequence means that for each positive integer \(n\text{,}\) \(p_n\lt p_{n+1}\text{.}\)

Definition 1.59

The statement that the sequence \(p_1,p_2,p_3,\dots\) is non-decreasing means that for each positive integer \(n\text{,}\) \(p_n\leq p_{n+1}\text{.}\)

Decreasing and non-increasing sequences are defined similarly.

Problem 1.61

Show that if \(M\) is a point set and \(p\) is a point and every closed interval containing \(p\) contains a point of \(M\) different from \(p\text{,}\)then \(p\) is a limit point of \(M\text{.}\)

Problem 1.62

Show that it is not true that if \(p\) is a limit point of a point set \(M\text{,}\) then every closed interval containing \(p\) must contain a point of \(M\) different from \(p\text{.}\)

Problem 1.63

(CA) True or false? If \([a,b]\) is a closed interval and \(G\) is a collection of open intervals with the property that every point in \([a,b]\) is in some open interval in \(G\) then there is a finite subcollection of \(G\) with the same property.

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