In-Depth Notes on Properties of Real Numbers and Cauchy Sequences

Properties of IR

  • IR, or the field of real numbers, is constructed as rational Cauchy sequences.

  • Theorem: IR is a field.

    • Definitions:

    • Let ( a = (an){n=1}^

    • 1 ) and ( b = (bn){n=1}^

    • 1 ) where ( an, bn \in \mathbb{R} ).

    • Both ( an ) and ( bn ) are Cauchy sequences.

Trichotomy in Cauchy Sequences

  • Given values of ( a ) and ( b ):

    1. If ( a > 0 ):

      • ( |an - bn| < \epsilon ) for all sufficiently large ( n ) implies ( ab > 0 ).

    2. If ( a = 0 ):

      • ( |an - bn| \leq |b_n| ) for all sufficiently large ( n ) implies ( a = b ).

    3. If ( a < 0 ):

      • The condition implies ( ab < 0 ) for sufficiently large ( n ).

Proof of Theorem

  • Let ( a \in R ): we need to show ( a ) has an inverse.

  • Write ( a = \frac{l{an}}{n+1} ) with ( ln ) being an upper bound sequence for ( an ).

Definition of Cauchy Sequence

  • A sequence ( (A_n) ) is a Cauchy sequence if for every ( \epsilon > 0), there exists ( N ) such that for all ( n,m > N ):

    • ( |an - am| < \epsilon )

Completeness Axiom of Reals

  • Every Cauchy sequence in ( IR ) converges.

  • For a given sequence ( a = (an) ), there is a rational number ( M > 0 ) such that ( an < M ) for all ( n ).

Density of Rationals

  • There exists a rational number ( q \in \mathbb{Q} ) such that ( q < b ).

Proof of Completeness of Reals

  • Let ( (a_n) ) be a Cauchy sequence of reals.

    • Choose ( \epsilon > 0 ) and for large ( n, m ), ensure ( |an - am| < \epsilon ).

Theorems on Limits and Bounds

  1. Least Upper Bound Property:

    • Every non-empty subset of ( R ) that is bounded above has a least upper bound.

    • Example: For set ( S ), there exists ( E ) as the least upper bound.

    • Proof: Define recursively upper bounds sequentially until convergence.

  2. Monotonic Convergence:

    • If ( (A_n) ) is monotonic and bounded above, it converges.

    • Proof: Let ( X ) be the least upper bound, then for any ( \epsilon > 0 ), there exists points converging to ( X ).

Continuous Functions

  • Proposition: A function ( f: X \to Y ) is continuous if:

    1. For every ( \epsilon>0 ), there exists ( \delta > 0 ) such that if ( dX(x, x') < \delta ), then ( dY(f(x), f(x')) < \epsilon ).

    2. Limits: If ( xn \to x ), then ( f(xn) \to f(x) ).

Intermediate Value Theorem

  • If ( f ) is continuous on ( [a, b] ) with ( f(a) < y < f(b) ), then there exists ( c \in (a, b) ) such that ( f(c) = y.\n

  • Proof: Defined intervals recursively to find point ( c ).

Exercises and Functions

  • To show that sum of continuous functions ( f \, , g ) is continuous,

    1. Prove by assuming the limit approach and properties of limits.

    2. Follow original definition of continuity and closures to intermix arguments.

Exponential Function

  • The exponential function is defined by ( exp(x) ).

IR (the field of real numbers) is constructed as rational Cauchy sequences, which are sequences where the terms become arbitrarily close to each other. The theorem states that IR is a field. A sequence (An) is a Cauchy sequence if for every (\epsilon > 0), there exists (N) such that for all (n, m > N), (|an - a_m| < \epsilon). Moreover, every Cauchy sequence in IR converges, ensuring completeness. The least upper bound property asserts that every non-empty subset of R that is bounded above has a least upper bound, while monotonic sequences that are bounded above converge. Continuous functions, defined by limits, adhere to properties ensuring continuity. For example, if function f is continuous on ([a, b]) with (f(a) < y < f(b)), there exists a point (c) in ((a, b)) where (f(c) = y). The exponential function is defined as (exp(x)).

IR (the field of real numbers) is constructed as rational Cauchy sequences, which are sequences where the terms become arbitrarily close to each other. The theorem states that IR is a field. A sequence (An) is a Cauchy sequence if for every (\epsilon > 0), there exists (N) such that for all (n, m > N), (|an - a_m| < \epsilon). Moreover, every Cauchy sequence in IR converges, ensuring completeness. The least upper bound property asserts that every non-empty subset of R that is bounded above has a least upper bound, while monotonic sequences that are bounded above converge. Continuous functions, defined by limits, adhere to properties ensuring continuity. For example, if function f is continuous on ([a, b]) with (f(a) < y < f(b)), there exists a point (c) in ((a, b)) where (f(c) = y). The exponential function is defined as (exp(x)).