\(\def\lim#1#2{ \underset{#1 \rightarrow #2}{lim} }\)

1 Preliminaries

1.1 Sets and Functions

• DEF: Function
• DEF: Injective
• DEF: Surjective

1.2 Boolean Logic

• DEF: Negation
• DEF: And
• DEF: Or
• DEF: Implication

• aVb = V(a,b) = T means a,b can take on the values of V^{-1}(T)
• Two functions are equal if
• a statement/function is equal to another statement with same domain when all variation of inputs gives same output
  • Can two statements that do not have the same input be equal to another?
• Truth table a b a->b
• Intersection of rows of truth table?
• a->b and !b->!a
  • (a,b) (F,T) (F,F) (T,T)
  • (!a,!b) (T,F) (T,T) (F,F) Almost same except for reversal of (T,F)

1.3

2 The Real Number

2.1 Algebraic and Order Properties of R

• DEF Operator \(+ *\) binary function from, to \(\mathbb{R}\)
• PRP \(a+b = b+a\)
• PRP \((a+b)+c = a+(b+c)\)
• PRP \(a*(b+c) = a*b + a*c\)
• DEF Special elements 0,1 wrt binary functions
  • \(a+0=a\), \(a*1=a\)
  • \(a+-a=0\), \(a*\dfrac{1}{a}=1\)
• PRP Uniquness of 0,1
• DEF Order \(a \leq b :=\) b-a is positive or zero
• PRP Reflexive \(a \leq a\)
• PRP Transitive \(a \leq b\) \(b \leq c\) \(\rightarrow a \leq c\)
• REM Negation \(b \leq a\)

• TFAE
• \(b \leq a\)
• a-b is positibe or zero
• b-a is negative or zero

2.2 ABS and Real-Line

• DEF \(|a| = \begin{cases} a > 0: a \\ a=0: 0 \\ a < 0 : -a \end{cases}\)

2.3 Completeness Property of R

• DEF Supremum/Infimum, least upper bound/greatest lower bound
• AXM upper bounded subset of \(\mathbb{R}\) has a least upper bound

2.4 Applications of Supremum

2.5 Intervals

3 Sequences

3.1 Sequences and Their Limits

• DEF Sequence
• DEF Limit

• The limit of sequence \((x_n)\) is \(L\) if: for any \(e > 0\),\(\exists k\) s.t. \(i > k \rightarrow |x_i-L| < e\)

• PRP Convergent sequence implies bounded
• DEF Negation of convergence to L

\(\exists e > 0\) s.t. \(\forall k \neg (i > k \rightarrow |x_i-L| < e)\) \(\exists e > 0\) s.t. \(\forall k ( \exists i > k) \land |x_i-L| > =e )\) There is an episilon, that no matter how far you go down in the sequence, there is an element in the sequence that lies outside the e neighborhood of L

3.2 Limit Theorems

• DEF Operators on sequences
• PRP Operator on sequences and their limits
• DEF Order of sequences
• PRP Order of sequences and their limits

3.3 Monotone Sequences

• PRP Monotone Bounded Sequences have a limit, the supremum

3.4 Subsequences and Bolzano Weierstrass Theorm

• DEF Subsequence
• PRP For a convergent sequence, every subsequence is convergent
• PRP Every sequence has a monotone subsequence

• The non existence of monotone subsequence implies the existence of one

3.5 Cauchy Criterion

• DEF Cauchy Sequence
• PRP sequence is cauchy <-> it is convergent
• PRP sequence is contractive -> it cauchy

3.6 Properly Divergent Sequences

• PRP sequence convergent -> bounded
• PRP sequence not bounded -> not convergent

3.7 Introduction to Series

• DEF Series of \((x_n)\) is a sequence \(s_n = x_1 + .. + x_n\)

• Sum within the sequence, product?

• PRP \(\sum{x_n}\) convergent \(\rightarrow\) \(Lim(x_n)=0\)
• PRP Series is convergent <-> series is bounded (if \((x_n)\) is positive)

• Series is convergent -> series is bounded
• Series is bounded. Series is monotone -> Series is convergent

• PRP Series, order and convergence

• Suppose \(0 = < x_n < y_n\). Then the following is true
• \(\sum{y_n}\) convergent \(- > \) \(\sum{x_n}\) convergent
• \(\sum{x_n}\) divergent \(- > \) \(\sum{y_n}\) divergent

• PRP Series, ratio and convergence

4 Limits on Functions

4.1 Limits of Functions

• DEF Limit of a function

• PRP: TODO \((a_n)\), \((b_n)\) cov to \(x\) then \(Lim(f(a_n)) = Lim(f(b_n))\) given this we can define uniqueness for anyseq
• DEF: \(Lim_{x \rightarrow a}(f(x)) = Lim( f(x_n) )\) where \(x_n\) is any sequence that conv to x
• Do we really need the sequence? Can we define it in an alternate way?

4.2 Limit Theorems

• DEF Operator on functions
• PRP Operator on functions and their limits

4.3 Some Extensions of the Limit Concept

• Different types of sequences from left,right
• Is there some property regarding the various limits of our function

• Combination of limits \(\lim{x}{a} g(f(x))\)

• DEF Continuous \( \lim{x}{a} f(x) = lim f(a) \)

• ??? Why is Continuity important (it allows us to define derivative)
• DEF \(\lim{x}{\infty} f(x)\)
• DEF \(\lim{x}{a} f(x) = \infty\)
• ??? What is derivative? Why important?

5.1 Continuous Functions 120

5.2 Combinations of Continuous Functions 125

5.3 Continuous Functions on Intervals 129

5.4 Uniform Continuity 136

5.5 Continuity and Gauges 145

5.6 Monotone and Inverse Functions 149