Theorem 12.6 Let {x.};= , and {y,},= , be convergent sequences of vectors in

R” with limits x and y, respectively; and let {c,,},=, be a convergent sequence

of real numbers with limit c. Then the sequence (c,x, + y.};=, converges to

limit cx + y.

Proof As usual, begin by choosing a positive number F. Note that

I Il(r,,x,, + y,,) - (cx + y)ll 5 IIc,J, all + lly. - yll. (6)

) Since the sequence of y.˜s converges to y, we know there exists an integer N,

such that for all II 2 Ni. IIy,, - y/I < c/2. On the other hand, for each com-

ponent i the sequence {cJ;,,} converges to c.xi by Theorem 12.3. By Theorem

;. 5 thl\ Imphe\: that the sequence (c,,x,,}˜˜˜ converges to cx. Thus there exists

.,I-

12

, an Nz such that for all n 2 N:, IIc˜,x,, - cxll < e/2. It follows that for all

PI 2_ N = max{N,, IV:}, llc,,x,, - cxJJ + lly,, yl\ i E and therefore by (6)

n

I Il(C,,X,, + y,,) (cx + y)ll < F.

A similar argument can he used to show that the sequence of inner products of

two convergent sequences of vectors converges to the inner product of the limits.

This is left as an exercise.

The definitions of accumulation point and subsequence extend naturally into

U”.

Definition The vector x is an accumulation point of the sequence {x,˜},:=, if

for any given F > (1 there are infinitely many integers n such that (lx. - XII < E.

A point that is a limit also satisfies the definition of an accumulation point-an

infinite number of terms in the sequence are within any given distance of the limit

point. But the definition of limit requires something more. All terms sufficiently

far out in a convergent sequence arc required to he within a given distence of the

limit, not just an infinite number of them. As we saw in R™, a consequence of

this distinction is that although a sequence can have several accumulation points,

a convergent sequence can have only one limit. The uniqueness of limits in Rm

follows directly from Theorems 12.1 and 12.5.

LlMlTS AND OPEN SETS [12]

264

Definition A sequence {Y˜};=˜ of vectors in R”™ is a subsequence of sequence

{xi}:=, in R”™ if there exists an infinite increasing set of natural numbers {nj) with

such that y, = x ,,,/ yz = x.., y3 = xl,>, and so on.

If a sequence has an accumulation point, it may nonethelw have no limit, as

d in Example 12.1 illustrate. However, each sequence with an

sequences c and

accumulation point has a subsequence that converges to one of its accumulation

points. For example, in scqucnce c the even-numbered terms form a convergent

d, the cvcn-numbered terms are a conver-

subsequence with limit 0. In sequence

gent subsequence with limit 1 while the odd-numhcrcd terms are a convergent

subsequence with limit I.

12.3 OPEN SETS

Our discussion of sequences leads naturally fu the study of open and closed sets

in R™“. Thr delinilion of a closed set directly requires the concept of a cunucrgcnt

sequence. The detinition of an open set requires the use of E-balls. Althou@

thcsc two definitions seem unrelated, we will xc that the concepts xc lruly

complementary.

WC start with open sets since they are the mast basic topological ctmslruction.

R”™ and a posilive

For a vector z in number E. the z-ball about z is R,.(z) =

{x t R”™ : IIx zII C: s). Somctimcs B,:(z) is called the open E-ball to distinguish

it from the closed ball {x E Km : IIx zll 5 z). which includes the boundary.

Open halls are important examples of a mnrc gcncral class of sets called open sets.

Definition Rm is open if for each x E 5™. there rxis& an open eball

A set S in

around x completely contained in S:

there is an tz > 0 such that R,:(x) C S.

XES -

an open neighborhood of x.

An open set S containing the point x is called

The word “open” has a connotation of .˜no boundary”: from any point one can

;dways move a little distance in url? direction and still he in the xt. The dclinition

I12.31 265

OPEN SETS

of an open set makes this idea precise: each element in an open set contains a

whole ball around it that lies in the set. Consequently, open sets cannot contain

their ˜*boundary points.”

Emmpk 12.4 The interval

(0, I) = {x t R : 0 < x < I}

is an open set. If h is a point in (0, l), then h # 0 or 1. The number h/2 is

closer to 0 than 0 is and is still in (0, l), while the number h + (1 b)/Z is

closer to 1 than b is and is still in (0, 1). If F = min{b/2, (1 b)/2}, the interval

(b F, b + E) is an open interval about b in (0, 1). (Check.)

The definition of an open set also implies that such sets are “thick” or “full-

dimensional,” since an open set in Rm contains an m-dimensional ball around each

of its points. Consequently, a line in RZ is not an open set. As indicated in Figure

12.4_ because a line is a one-dimensional subset of R2, the ball around any point

on the line contains points which arc not on the line. Similarly, a line or plane in

R” cannot he open and one-point sets are never open.

1 Theorem 12.7 Open halls are open sets.

Proof Let R bc the open ball B,(x) ahout x. and let y he an arbitrary point in b™.

We want to show that thcrc is some hall ahout y that lies completely in B. Let

that the open ball V of radius tz S around y

Figure 12.5. Let z be an arbitrary point in V.

266 UMITSANDOPENSETS 1121

Then, by the triangle inequality,

112 ˜ XII s (12 - y]I + Ily - XII < (E 6) + 6 = E.

Thus. V C B. W

Figure

/fy E B,(x) and 6 = IIx ˜ yll, then B,-*(y) C B,(x).

12.5

The next theorem describes the behavior of open sets under the set operations

of union and intersection and allows us to construct interesting examples of open

sets.

Theorem 12.8

(a) Any union of open sets is open.

(h) Thefinite intersection of open sets is open.