Chapter 1 The Real Numbers: Sets, Sequences, and Fuctions
Measure : Some examples
• Let C be a curve in R2 with the vector function r(t) =< x(t), y(t) > for t ∈ [a, b]. The length of C is defined by
L(C) = Z
C
ds = Z b
a
r dx dt
2
+ dy dt
2
dt, i.e., we measure the leng of C.
• Let f be a continuous function on R2. Then we estimate Lf(C) =
Z
C
f (x, y)ds, that is, the ‘line integral’ of f along the curve C.
• How to measure [0, 1] ∩ Q?
Sets
Definition. A set E in R is dense in R if for any a, b ∈ R (a < b), there is c ∈ E such that a < c < b.
Theorem 1. The set of rational numbers Q is dense in R.
Equipotent sets: We call two sets A and B equipotent if there is a one-to-one mapping f from A and B, denoted by A ∼= B.
Definition Let Nn= {1, 2, . . . , n} ⊂ N. For any set A, we say:
(1) A set E is finite if E is empty, i.e., E = φ or it has only a finite number of elements in E. That is, there is a number n for which E is equipotent to Nn.
(2) E is infinite if E is NOT finite.
(3) E is countably infinite if E ∼= N. Countable sets are sometimes called enumerable or denumerable sets.
(4) E is (at most) countable if E is finite or countable.
(5) E is uncountable if E is neither finite nor countable.
Theorem 2. The following sets are countable.
(1) N × N × · · · × N
(2) The set of rational numbers Q.
(3) The union of countable collection of countable sets. That is, if {En} be a sequence of countabe sets, then E = ∪∞n=1{En} is countable.
Theorem 3. The sets (a, b), [a, b) and [a, b] are uncountable.
Sequence of Real Numbers
Sequence : A sequence of real numbers is a function on N to R a : N 3 n 7−→ a(n) =: an.
Limit point : Let (an) be a sequence of real numbers. Then, a is called a limit of (an) i.e.,
n→∞liman= a, if for any > 0, there exists an integer N > 0 such that
|an− a| < , ∀n ≥ N.
Let a be the limit of (an). Then, for any > 0, all but a finite number of an are in the ball B(x, ).
Theorem 4. (Bolzano-Weirstrass Theorem) Every bounded sequence of real numbers has a convergent subsequence.
Theorem 5. A subset E of R is closed if and only if it contains all its limit points.
Theorem 6. A monotone sequence converges if and only if it is bounded.
Cauchy Sequence: A sequence (an) is called Cauchy sequence if for any > 0, there exists an integer N > 0 such that
|an− am| < , ∀n, m ≥ N.
Theorem 7. A sequence of real numbers converges if and only if it is a Cauchy sequence.
Cluster Point: Let (an) be a sequence of real numbers. Then, a is called a cluster point of (an) if for any > 0 and N > 0, there exists an integer n ≥ N such that
|an− a| < .
For instance, consider an= (−1)n.
limsup/inf: Let (an) be a sequence of real numbers. Define liman= inf
n sup
k≥n
ak= lim sup an.
Then, ` = lim sup an iff (i) For any > 0, there exists n ∈ N such that ak< ` + for all k ≥ n and (ii) given > 0, and n, there exists k ≥ n such that ak > ` − .
Similarly, we can define
liman= sup
n
k≥ninf ak= lim inf an. Properties: Let (an) be a sequence of real numbers. Then
(1) lim (−an) = − lim an. (2) lim an ≤ lim an.
(3) If an→ `, then ` = lim an= lim an.
Open and Closed Sets of Real Numbers Properties:
(1) The intersection of any finite collection of open sets is open.
(2) Every open set of real numbers is the union of a countable collection of disjoint open intervals.
(3) The union of any collection of open sets is open
(4) The union of any finite collection of closed sets is closed.
(5) The intersection of any collection of closed sets is closed.
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Theorem 8. (Heine-Borel) Let F be a closed and bounded set in R. Then, each open covering of F has a finite subcovering.
Continuity: A real-valued function f defined on E is said to continuous at x ∈ E if given
> 0, there exists δ > 0 such that for all y with |x − y| < δ, we have
|f (x) − f (y)| < .
Uniform Continuity: A real-valued function f definedon E is said to uniformly continu- ous (on E) if given > 0, ∃δ > 0 such that for all x, y with |x − y| < δ, we have
|f (x) − f (y)| < .
A continuous function on a closed and bounded set is uniformly continuous.
Lipschitz Continuity: A real-valued function f defined on E is said to be Lipschitz (on E) if ∃c > 0 such that for all x, y in E.
|f (x) − f (y)| ≤ c|x − y|
Theorem 9. (Extreme Value Theorem) A continuous function on a closed and boundned set takes a minimum and a maximum values
Theorem 10. (Intermediate Value Theorem) Let f be a continuous function on [a, b] and f (a) < c < f (b). Then, there is number x0 ∈ (a, b) such that f (x0) = c. set takes a minimum and a maximum values
Pointwise Convergence: Let (fn) be a sequence of function defined on E. Then (fn) is said to converge pointwise if for every x ∈ E, we have f (x) = lim fn(x) : that is, if given x ∈ E and > 0, ∃N such that
|f (x) − fn(x)| < ∀n ≥ N.
Uniform Convergence: A sequence (fn) of function defined on E is said to converge uni- formly if given > 0, ∃N such that
|f (x) − fn(x)| < ∀x ∈ E, n ≥ N.
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