Heine-borel Theorem. Bolzano-Weierstrass Theorem.

Answer:

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 S_t=S [α;t] for ≥α.

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

Since S_β =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, S_αis the singleton set {α}, which is contained in some open set H_αform 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 S_λ-ε=S_λ+ε

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

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

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

Since S_β-ε is covered by a finite collection of set {H₁,…,H_k}, S_p is covered by the finite collection {H₁,…,H_k,H_β}. Since S_β=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 N_x that contains no point of other than x. The collection H={N_x| 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 N_x1,…,N_xn. Since these set contain only x₁,x₂,…x_n from S, it follows that S={x₁,…,x_n}

 

Sequences of real numbers. Monotonic Sequences.

Answer:

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

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

In this case we say that {x_n} 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 |x_n-a|<ε if n>N₁ (because _n=a); |x_n-b|<ε if n>N₂ (because _n=b)

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

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

Monotonic Sequences.

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

Theorem. If a sequence {x_n} is monotonic and bounded then it is convergent. Cauch’s Convergence Criterion. A sequence {x_n} converges inӀR_n if and only if for every ε>0 there is n_εͼN, for every n>n_ε,for every pͼN => |x_n+p-x_n|<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 t_n 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 {S_n} and denoted by Ṡ= _n and S = _n

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