2: Sequences
- Page ID
- 49105
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We introduce the notion of limit first through sequences. As mentioned in Chapter 1, a sequence is just a function with domain \(\mathbb{N}\). More precisely, a sequence of elements of a set \(A\) is a function \(f: \mathbb{N} \rightarrow A\). We will denote the image of \(n\) under the function with subscripted variables, for example, \(a_{n}=f(n)\). We will also denote sequences by \(\left\{a_{n}\right\}_{n=1}^{\infty}\), \(\left\{a_{n}\right\}_{n}\), or even \(\left\{a_{n}\right\}\). Each value \(a_{n}\) is called a term of the sequence, more precisely, the \(n\)-th term of the sequence.
Consider the sequence \(a_{n}=\frac{1}{n}\) for \(n \in \mathbb{N}\).
Solution
This is a sequence of rational numbers. On occasion, when the pattern is clear, we may list the terms explicitly as in
\[(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots \nonumber\]
Let \(a_{n}=(-1)^{n}\) for \(n \in \mathbb{N}\). This is a sequence of integers, namely,
\[-1,1,-1,1,-1,1, \ldots \nonumber\]
Solution
Note that the sequence takes on only two values. This should not be confused with the two-element set \(\{1,-1\}\).
- 2.1: Convergence
- This page covers the concept of convergence for sequences of real numbers, defining key terms and presenting formal definitions. It discusses conditions for convergence, the uniqueness of limits, and introduces important theorems such as the Comparison Theorem and the Squeeze Theorem. The behavior of subsequences is also explored, highlighting that they converge to the same limit as the original sequence if it converges.
- 2.2: Limit Theorems
- This page covers theorems on computing limits of sequences, highlighting that the convergence of two sequences allows for predictable limits in their combinations. It includes proofs, estimates, and examples that clarify these concepts, particularly the limit as \(n\) approaches infinity. Exercises are provided to reinforce understanding, emphasizing the significance of convergence in sequences and including challenges on proving statements about sums and products.
- 2.3: Monotone Sequences
- This page covers key concepts of monotone sequences, including definitions of increasing and decreasing sequences, the Monotone Convergence Theorem, and properties of limits. It explains how bounded increasing sequences converge to a limit (e.g., \(e\)) and distinguishes between divergent sequences. The behavior of sequences under various operations is addressed, with Theorems 6 and 7 detailing limit properties and the comparison theorem.
- 2.4: The Bolazno-Weierstrass Theorem
- This page covers the Bolzano-Weierstrass Theorem, which states that any bounded sequence of real numbers has a convergent subsequence, linked to the completeness axiom. It includes a proof using nested intervals and discusses Cauchy sequences, asserting their convergence. It also defines contractive sequences, highlighting that they are convergent under specific conditions.
- 2.5: Limit Superior and Limit Inferior
- This page provides insights into bounded sequences and the convergence of their supremum and infimum. It defines limit superior and limit inferior, clarifying their relationship with sequence behavior. Key theorems state that convergence occurs when the limit inferior and superior align, while subsequential limits impact bounded sequences.
- 2.6: Open Sets, Closed Sets, Compact Sets, and Limit Points
- This page covers key concepts in real analysis, including open and closed sets in \(\mathbb{R}\), compactness, boundedness, and convergence. It explains that an open set contains intervals around points, while a closed set has an open complement. The definition of compactness involves sequences converging within the set. Theorems clarify properties of open/closed sets, including the significance of the empty set and \(\mathbb{R}\) itself.