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EXERCISES

12.1 Write out the nth term for the rest of the sequences in Example 12.1.

12.2 Explain why each of the following se& is not a subsequence of the last sequence in

Example 12.2:

12.3 Give a direct proof of inequalities (2) and (3) for real numbers

260 LIMITS AND OPEN SETS [I 21

12.4 Prove that lxyl = 1x1 lyl for all x, y.

12.5 prove that lx + y + 11 5 1x1 + IyI + Izl for all numbers x, y, z.

12.6 Prove that if {x,}, , and J&J,= I are sequences with limits x and y, respectively, then

the sequence (x. y,,},= L converges to the limit x - y.

Suppose that {I& is B sequence of real numbers that converges to x0 and that all

12.7

xn and x0 are nonzero.

a) Prove that there is a positive number B such that lx,\ 2 B for all n.

b) Using a, prove that {l/x,,} converges to {I /x01.

Let {x,};-, and b,,),=, be convergent sequences with limits x and y, respectively.

12.8

Suppose that all the y,,вЂ˜s and y are nonzcro. Show that the sequence {xX./y,},,

converges to x/y.

A sequence is said to be bounded if there is a number B such that lx,,1 5 B for all

12.9

n. Show that if {x,},=, converges to 0 and if bn},-] is bounded, then the product

sequence converges to 0.

12.10 Write out the proof of the last sentence in the statement of Theorem 12.4.

12.2 SEQUENCES IN Rm

A sequence in RвЂќвЂ™ is just what we would expect it to be: an assignment of a vecfor

in RвЂќвЂ™ to each natural number n: {x,, x2, ˜3,. .}. For such sequences we need to

keep track of two different indices: one for the m coordinates of each m-vector,

and the other to indicate which element in the sequence is under consideration.

Before we can carefully define convergence. we need to make precise our

notion of closeness in Rm, as we did at the beginning of the previous section for

sequences in RвЂ™. Recall from Chapter IO that the distance between any two vectors

x and y in Rm is the norm of their difference:

d(x, y) = I/x - ytt = &I - y,)вЂ™ + + (xn, - Y,)вЂ˜.

By this definition. the distance between two numbers is the length of the line

segment between them, as indicated in Figure 12.2. The distance mcasurc or

Figure

12.2

I1 2.21 SEOUENCES Rm 261

IN

вЂњmetricвЂќ defined by the Euclidean norm of the difference is frequently referred to

as the Euclidean metric.

The triangle inequality (Theorem 10.5) implies that for any three vectors x, y,

and z,

IIX - zll = II@ - y) + (y - z)ll

5 IIX - YII + IIY -A,

вЂњT d(x, 2) 5 d(x, Y) + d(y, 2). (4)

The generalization of the einterval Ia about a point r on RвЂ™ is the &-ball

iTI Rm.

Definition Let r be a vecta in Rm and let E bc a positive number. The e-ball

about r is

B,(r) = {x E RвЂќвЂ™ : IIx ˜ rll < E}.

Intuitively, a vector x in Rm is close to r if x is in some B,(r) for a small but

positive E. The smaller E is, the closer x is to r.

Definition Ascquence of vcctors{x,, x7, x3,. .} is said to converge to thevcctor

x if for any choice of a positive real number F, there is an integer N such that. for

all II 2 N. x,, t B,(x): that is,

d(x,,. x) = IIx,, ˜ XII < e.

The vector x is called the limit of the sequence.

In other words. a sequence of vectors {x,,};=, converges to a limit vectw x

if and only if the sequcncc of distances from the vector x,, to x. {/Ix,, - XII},-,,

converges to 0 in RвЂ™.

Exomplr IL.1 Because of the extra dimensions tu move around in, a convergent

seqocnce can mwc in all kinds of spirals in RвЂќвЂ™ as it converges to its limit. For

example. let {u,,} be the sequence of + Is and -Is which changes sign euer˜

othcv ,<wn:

{ll,,},m, = {I, I. -˜I. -I, 1. I, -I, -1, 1. I, PI, -1, l,... }.

I

u,+I

Nowl construct the sequence {x.},=, in which x,, = ( вЂќ ,---):

n (n+l)

Note that the x,,вЂ˜s move clockwise in R2 from quadrant to quadrant. as they

convcrgc to the origin, as illustrated in Figure 12.3.

L,M,TS AND OPEN SETS [I 21

262

Figure

12.3

However, if the sequence {x,,} is converging to x, then each component of the x,,вЂ˜s

must be conver@ng tu the corresponding component of the limit vector x, and

conversely. This equivalence hetwccn the convergence of a sequence of vectors

and the convergence of each of its components reduces the prohlcm of verifying

the convcrgcncc of a sequence of vectors in Rm to the problem of verifying the

convergence of M scqucnces in RвЂ™.

converges if and only if all m

Proof (If) Let {x,,};-, he a sequence ofvectors in RвЂќвЂ˜. Write x,, = (xl,,, , x,,,,,).

Suppose each of the nz sequences of numbers {xi,,},-, , i = I, , m. convcrgcs

to a limit .xi;. I.et xвЂќ = (XT.. A-;;,). Choose and tix a small positive number E.

For each i between I and m, there exists an intcgcr N; such that. if II 2 N,, then

/x0, - .?˜;вЂ˜I < a/,/%. L e t N = max{N,, Nr,,). Supposc II 2 N . Then.

112.21 SEQIJENCES tzm 263

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/Ix, - x*11 < F. But then for n P N and for each component i,

lXin - $1 c (Xln xy + + (Xm,, - $J*

= (lx,, - x*11 i E. n

Theorem 12.5 enables us to apply the results about sequences in RвЂ™ to se-

quences in RвЂќвЂ˜. For example, the next theorem is the generalization of Theorems

12.2 and 12.3 in that it states that all the vector operations we have defined in

previous chapters are preserved under limits

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