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5.1: CHAPTER 1
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/05%3A_Solutions_and_Hints_for_Selected_Exercises/5.01%3A_Section_1-
Using $frac{\varepsilon}{2}$ in part $(2^{\prime})$ of Proposition 1.5.1 applied to the sets $A$ and $B$ , there exits $a \in A$ and $b \in B$ such that \[\sup A-\frac{\varepsilon}{2}<a \te...Using $frac{\varepsilon}{2}$ in part $(2^{\prime})$ of Proposition 1.5.1 applied to the sets $A$ and $B$ , there exits $a \in A$ and $b \in B$ such that $\sup A-\frac{\varepsilon}{2}<a \text { and } \sup B-\frac{\varepsilon}{2}<b .$ It follows that $\sup A+\sup B-\varepsilon<a+b .$ This proves condition $(2^{\prime})$ of Proposition 1.5.1 applied to the set $A + B$ that $\sup A + \sup B = \sup (A + B)$ as desired
5.2: CHAPTER 2
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/05%3A_Solutions_and_Hints_for_Selected_Exercises/5.02%3A_Section_2-
Define $\alpha_{n}=\sup _{k \geq n}\left(a_{n}+b_{n}\right), \beta_{n}=\sup _{k \geq n} a_{k}, \gamma_{n}=\sup _{k \geq n} b_{k} .$ By the definition, \[\limsup _{n \rightarrow \infty}\left(a_{n}+b_...Define $\alpha_{n}=\sup _{k \geq n}\left(a_{n}+b_{n}\right), \beta_{n}=\sup _{k \geq n} a_{k}, \gamma_{n}=\sup _{k \geq n} b_{k} .$ By the definition, $\limsup _{n \rightarrow \infty}\left(a_{n}+b_{n}\right)=\lim _{n \rightarrow \infty} \alpha_{n}, \limsup _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \beta_{n}, \limsup _{n \rightarrow \infty} b_{n}=\lim _{n \rightarrow \infty} \gamma_{n} .$ By Exercise 2.5.3, \[\alpha_{n} \leq \beta_{n}+\gamma_{n} \text { for all } n \in \math…
4.5: Taylor's Theorem
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/04%3A_Differentiation/4.05%3A_Section_5-
Consider the function $g(t)=f(x)-\sum_{k=0}^{n} \frac{f^{(k)}(t)}{k !}(x-t)^{k}-\frac{\lambda}{(n+1) !}(x-t)^{n+1} .$ Then \[g(\bar{x})=f(x)-\sum_{k=0}^{n} \frac{f^{(k)}(\bar{x})}{k !}(x-\bar{x})^{k...Consider the function $g(t)=f(x)-\sum_{k=0}^{n} \frac{f^{(k)}(t)}{k !}(x-t)^{k}-\frac{\lambda}{(n+1) !}(x-t)^{n+1} .$ Then $g(\bar{x})=f(x)-\sum_{k=0}^{n} \frac{f^{(k)}(\bar{x})}{k !}(x-\bar{x})^{k}-\frac{\lambda}{(n+1) !}(x-\bar{x})^{n+1}=f(x)-P_{n}(x)-\frac{\lambda}{(n+1) !}(x-\bar{x})^{n+1}=0 .$ and $g(x)=f(x)-\sum_{k=0}^{n} \frac{f^{(k)}(x)}{k !}(x-x)^{k}-\frac{\lambda}{(n+1) !}(x-x)^{n+1}=f(x)-f(x)=0 .$ By Rolle's theorem, there exists $c$ in between $\bar{x}$ and $x$ such that…
2: Sequences
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/02%3A_Sequences
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_{...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.
Back Matter
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/zz%3A_Back_Matter
4.1: Definition and Basic Properties of the Derivative
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/04%3A_Differentiation/4.01%3A_Definition_and_Basic_Properties_of_the_Derivative
For example, given functions $f: G_{1} \rightarrow \mathbb{R}$ , $g: G_{2} \rightarrow \mathbb{R}$ , and $h: G_{3} \rightarrow \mathbb{R}$ such that $f\left(G_{1}\right) \subset G_{2}$ , \(g\left...For example, given functions $f: G_{1} \rightarrow \mathbb{R}$ , $g: G_{2} \rightarrow \mathbb{R}$ , and $h: G_{3} \rightarrow \mathbb{R}$ such that $f\left(G_{1}\right) \subset G_{2}$ , $g\left(G_{2}\right) \subset G_{3}$ , $f$ is differentiable at $a$ , $g$ is differentiable at $f(a)$ , and $h$ is differentiable at $g(f(a))$ , we obtain that $h \circ g \circ f$ is differentiable at $a$ and $(h \circ g \circ f)^{\prime}(a)=h^{\prime}(g(f(a))) g^{\prime}(f(a)) f^{\prime}(a)$
5: Solutions and Hints for Selected Exercises
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/05%3A_Solutions_and_Hints_for_Selected_Exercises
4: Differentiation
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/04%3A_Differentiation
In this chapter, we discuss basic properties of the derivative of a function and several major theorems, including the Mean Value Theorem and l'Hôpital's Rule.
3: Limits and Continuity
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/03%3A_Limits_and_Continuity
In this chapter, we extend our analysis of limit processes to functions and give the precise definition of continuous function. We derive rigorously two fundamental theorems about continuous functions...In this chapter, we extend our analysis of limit processes to functions and give the precise definition of continuous function. We derive rigorously two fundamental theorems about continuous functions: the extreme value theorem and the intermediate value theorem.
1: Tools for Analysis
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/01%3A_Tools_for_Analysis
This chapter discusses various mathematical concepts and constructions which are central to the study of the many fundamental results in analysis. Generalities are kept to a minimum in order to move q...This chapter discusses various mathematical concepts and constructions which are central to the study of the many fundamental results in analysis. Generalities are kept to a minimum in order to move quickly to the heart of analysis: the structure of the real number system and the notion of limit. The reader should consult the bibliographical references for more details.
3.1: Limits of Functions
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Mathematical_Analysis_I_(Lafferriere_Lafferriere_and_Nguyen)/03%3A_Limits_and_Continuity/3.01%3A_Limits_of_Functions
Then $f$ does not have a limit at $\bar{x}$ if and only if there exists a sequence $\left\{x_{n}\right\}$ in $D$ such that $x_{n} \neq \bar{x}$ for every $n$ , $\left\{x_{n}\right\}$ conv...Then $f$ does not have a limit at $\bar{x}$ if and only if there exists a sequence $\left\{x_{n}\right\}$ in $D$ such that $x_{n} \neq \bar{x}$ for every $n$ , $\left\{x_{n}\right\}$ converges to $\bar{x}$ , and $\left\{f\left(x_{n}\right)\right\}$ does not converge. Let $\left\{x_{n}\right\}$ be a sequence in $B(\bar{x} ; \delta) \cap D=(\bar{x}-\delta, \bar{x}+\delta) \cap D$ that converges to $\bar{x}$ and $x_{n} \neq \bar{x}$ for all $n$ .

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