Course: CALCULUS 1

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity Announcements
  
  Announcements Forum
Select section 1. The Real Number System

Collapse Expand
1. The Real Number System
- Collapse Expand
  1.1 Algebraic Properties of R
  
  This section introduces the foundational structure of the real numbers as a complete ordered field.
  
  Content:
  
  Field axioms (addition and multiplication properties)
  
  Order axioms (trichotomy law, transitivity, etc.)
  
  The interaction between algebraic and order structures
  
  The absolute value function and its properties
  
  Intervals: open, closed, half-open, bounded/unbounded
  
  Learning Outcomes:
  
  Understand and apply the axioms of a field and an ordered set
  
  Use inequalities and algebraic manipulation within ℝ
  
  Interpret and construct subsets of ℝ using interval notation
  
  Select activity The Field Axioms of ℝ Notes
  
  The Field Axioms of ℝ Notes Book
  
  Select activity The Order Axioms of ℝ Notes
  
  The Order Axioms of ℝ Notes Book
  
  Select activity Downloadable PDF notes on field axioms
  
  Downloadable PDF notes on field axioms File PDF
  
  Select activity nptel course on analysis
  
  nptel course on analysis URL
Select section Sequences and Limits

Collapse Expand
Sequences and Limits
- Collapse Expand
  New subsection
  
  Sequences in Real Analysis A sequence in real analysis is a function from the set of natural numbers into the real numbers, typically denoted as ( 𝑎 𝑛 ) 𝑛 = 1 ∞ (a n ) n=1 ∞ , where each term 𝑎 𝑛 a n represents the value of the sequence at the 𝑛 n-th position. Sequences are fundamental to understanding limits, convergence, and the structure of the real number system. Key topics include bounded and monotonic sequences, subsequences, limit points, and the Bolzano-Weierstrass Theorem. Sequences provide the groundwork for more advanced concepts such as series, continuity, and differentiability.
  
  Select activity SEQUENCES
  
  SEQUENCES Database
  
  Opened: Monday, 21 July 2025, 2:59 PM
  
  Select activity What are sequences and series
  
  What are sequences and series URL
- Collapse Expand
  Limits of a sequence
  
  A sequence
  $\{a_n\}$
  {an} converges to a real number
  $L$
  L if:
  $\forall \epsilon > 0, \exists N \in \mathbb{N} \text{ such that } n > N \Rightarrow |a_n - L| < \epsilon$
  $\forall ϵ > 0, \exists N \in N such that n > N \Rightarrow ∣ a n - L ∣ < ϵ$
  
  We write
  $\lim_{n \to \infty} a_n = L$
  $lim n \to \infty a n = L$ .
  If no such
  $L$
  L exists, the sequence diverges.
  
  Basic limit theorems:
  
  $\lim (a_n \pm b_n) = \lim a_n \pm \lim b_n$
  $lim (a n \pm b n) = lim a n \pm lim b n$
  
  $\lim (a_n b_n) = (\lim a_n)(\lim b_n)$
  $lim (a n b n) = (lim a n) (lim b n)$
  
  $\lim \left( \frac{a_n}{b_n} \right) = \frac{\lim a_n}{\lim b_n}$
  $lim (b n a n) = l i m b n l i m a n$ , provided
  $\lim b_n \neq 0$
  $lim b n \neq= 0$
Select section Cauchy Sequences

Collapse Expand
Cauchy Sequences
Select section Series of Real Numbers

Collapse Expand
Series of Real Numbers
Select section Topology of ℝ

Collapse Expand
Topology of ℝ
Select section Limits of Functions

Collapse Expand
Limits of Functions
Select section Continuity

Collapse Expand
Continuity
Select section Differentiation

Collapse Expand
Differentiation
Select section The Riemann Integral

Collapse Expand
The Riemann Integral
Select section Sequences and Series of Functions

Collapse Expand
Sequences and Series of Functions