Heine-borel Theorem. Bolzano-Weierstrass Theorem.


If H is an open covering of a closed and bounded subset S of the real line, then S has an open covering H consisting of finitely many open sets belonging to H.

Proof: Since S bounded, it has an infimumα and supremum β , and, since S is closed,α and β belong to S. Define St=S [α;t] for ≥α.

and let F={t|α<t<β and finitely many sets form H cover St}

Since Sβ =S1 the theorem will be proved, if we can show that β F. To do this, we use the completeness of the reals.

Since α S, Sαis the singleton set {α}, which is contained in some open set Hαform H, because H covers S; therefore, αcF.

Since, F is nonempty and bounded above by β, it has supremumλ. First, we wish to show that λ=β by definition of F , it suffices to rull out the possibility that α<β.

We consider two cases:

Case 1. Suppose that α<β and λ S. Then, since S is closed λ is not a limitpointof S. Consequently, there is an ε>0 such that [λ-ε, λ+ε] S= , so Sλ=Sλ+ε

However, the definition of λ implies that Sλ-ε has a finite subcovering from H, while Sλ+εdoes not. This is a contradiction.

3.2 Case 2.Suppose that λ<β and λ S. Then there is an open set Hλ in H that contains λ and along with λ, an interval [λ-ε, λ+ε] for some positive ε. Since Sλ-ε has a finite covering.{H1,…,Hn} of sets from H, it follows, that Sλ+ε has the finite covering{H1,….,Hn,Hλ}. This contradicts the definition of λ.

Now we know that α=β, which is in S. Therefore, there is an open set Hβ in H that contains β and along with, an interval of the from [β-ε,β+ε], for some positive ε.

Since Sβ-ε is covered by a finite collection of set {H1,…,Hk}, Sp is covered by the finite collection {H1,…,Hk,Hβ}. Since Sβ=S1 .

Henceforth , we will say a closed and bounded set is compact.

As an application of the Heine-Borel theorem, we prove the following theorem of Bolzano andWeierstrass.

Bolzano-Weierstrass theorem. Every boumded infinite set of real numbers has at least one limit point.

Proof: We will show that a bounded nonempty set without a limit point can contain only a finite number of points.

If S has no limitpoints,then S is closed and every point x of S has an open neighborhood Nx that contains no point of other than x. The collection H={Nx| x S} is an open covering for S. Since S is also bounded Heine-Borel theorem implies that S can be covered by a finite collection of sets from H1,…. say Nx1,…,Nxn. Since these set contain only x1,x2,…xn from S, it follows that S={x1,…,xn}


Sequences of real numbers. Monotonic Sequences.


We denote a sequence by {xn}oon=1. The real number xn is the n-th term of the sequence.

Definition. A sequence {xn} converges to a limit a if for every ε>0 there is an integer nε such that |xn-a|<ε, if n>nε

In this case we say that {xn} is convergent and we write n=a.

A sequence that does not converge diverges or is divergent.

Theorem. The limit of a convergent sequence is unique.

Proof: Suppose that n=a and n=b. We must show that a=b. Let ε>0. From definition of limits, there are integers N1 and N2 such that |xn-a|<ε if n>N1 (because n=a); |xn-b|<ε if n>N2 (because n=b)

This inequelities both hold, if n>N=max(N1, N2) which implies that |a-b|=|a-xn+xn-b|≤|a-xn|+|xn-b|=ε+ε=2ε

Since, this inequality holds for every ε>0 and |a-b|=0; that is a=b

We say that n= , if for any real number axn>a for large n

Definition. A sequence {xn} is bounded above if there is a real number b such that xn≤b for all n , bounded below, if there is a real number a such that xn≥a for all n , or bounded, if there is a real number r such that |xn|≤r (-r≤xn≤r) for all n

Monotonic Sequences.

Definition: A sequence {xn} is increasing, if xn<xn+1 ; non-decreasing, if xn≤xn+1 ; decreasing, if xn+1<xn; non-increasing, if xn+1≤xn.

Theorem. If a sequence {xn} is monotonic and bounded then it is convergent. Cauch’s Convergence Criterion. A sequence {xn} converges inӀRn if and only if for every ε>0 there is nεͼN, for every n>nε ,for every pͼN => |xn+p-xn|<ε.

Theorem. Let n=S and n=t where S and t are finite. Then:

1. n=cS , c is constant.

2. =S+t, (the limit of is the sum of the limits)

3. n=S-t,

4. =St,

5. = (if tn is non-zero for all n and t 0),(the limit of quotient is the quotient of the limits, provided t 0)

Definition. The numbers Ṡ and S are called the limit superior and limit inferior respectively of {Sn} and denoted by Ṡ= n and S= n

Theorem.If {an} is a sequence of real number, then n=b if and only if | n= n=b

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