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write F(n) as x,,. Now, let M be any infinite subset of the natural numbers. Write

M as {n,, n2, ni,. .}, where

Create a new sequence {y,J, where

= xn,, for j = 1, 2, 3,

yj

This new sequence bj};=, is called a subsequence of the original sequence (x,˜}. In

short, we construct a subsequence of a sequence by choosing an infinite collection

of the entries of the original sequence in the order that these elements appear in

the original sequence. (See Exercise 12.2.)

Algebraic Properties of limits

It should be intuitively clear that limits of sequences are preserved by algebraic

operations. For example. if (x.}:=, gets arbitrarily close to x and if {.y,,],T-, gets

112.11 SEQUENCESOF REAL NUMBERS 257

arbitrarily close toy, then the sequence of the sums {xn + yn}zzl gets arbitrarily

close to x + y, and similarly for differences, products, and quotients. Work at

understanding the next two proofs as conipletely as possible in order to develop

a good working knowledge of the notion of limits, and of the notion of a careful

proof. The proofs of most theorems about sequences and their limits require the

triangle inequality, often more than once:

Ix + Yl 5 1x1 + IYI for all x, y. (2)

Some proofs also call for the subtraction variant of the triangle inequality:

(3)

) 1x1 - lyl ) s lx - yl for all x, y.

For real numbers, one can prove inequalities (2) and (3) directly, for example,

by looking at cases based on the signs of x and y. (Exercise.) For vectors in

RвЂќ, one replaces the absolute values in inequality (2) by norms. In this case, the

corresponding statement is the Triangle Inequality. See Theorems 10.5 and 10.6.

Theorem 12.2 Let {I,˜},-˜ and b,,};=, be sequences with limits x and y.

respectively. Then the sequence {x, + y,},, converges to the limit x + y.

Proof Choose and tix a small positive number E. [Just about all proofs about

1limits begin this way.1 Since we know that x,, -+ x and y,, - y, there exists an

˜integer N, such that

and an integer IVY such that

вЂ™ ly,, y/ < ; fвЂќr n 2 Nz

1Let N = max{N,, N2}. Then for all n 2 N,

I(& + y.) - 6 + L.)I = l(G - x) + (yo y)l

5 Ix, -xl + lyn yl by (2)

n

E.

I

We next prove that if xn - x and y,? - y. then x,+. - xy. This proof is a

bit more tedious than that of Theorem 12.2, and can bc skipped at first reading.

We will begin by writing the critical difference Ixy x,,y,l as a sum of three

LIMITS AND OPEN SETS [I 21

258

terms, each of which involves the distances lx,, - XI and ly, - 1.1 which we know

something about. We will then choose a positive E. To show that the sum of the

three terms in less than 8 for larg&˜N, we show that each is less than z/3 for large

enough N.

Theorem 12.3 Let {&}r==, and b8,}z=, be sequences with limits x and y,

respectively. Then the sequence of products {,Y˜x,Y,},, converges to the limit

XY.

Proof To show that Ixy x,,y,l is small when Ix ˜ x.1 and Iy y,,l are small,

try to write the former in terms of the latter two. We will accomplish this by

using the mathematicianвЂ™s trick of adding and subtracting the same element to

a given expression: in fact, weвЂ™ll do this twice.

Ix - x,2 Y,,I = Ix Y - x Y,, + x Ysz n,, Ynl

Y (once)

= Ix (Y - Yn) + (x - x,x) Ynl

= Ix (Y ˜ Y.) + (x 4,)˜ (YвЂќ - Y + Y)I (twice)

= Ix (Y - Y,,) + (x - 4 (Y,, Y) + (x ˜ &J YI

5 lx (Y - ?d + 1(-r - &z) (Y,z - Y)I + lb ˜ 4 4вЂ™1

(using the triple triangle inequality)

5 IXIIY ˜ YJ + Ih. X,,llY,, Y/ + I+ вЂњ.l/Yl.

We know that each term in the last expression goes to zero. To make this pmccss

precise, proceed just as in the proof of Theorem 12.2. Choose and fix a small

positive numhcr c, with .? < I. Since x,, -I, there is an integer N, such that

n?N, =+

In - &,I < 3(ly;+ 1)

Since y,, - y: there is an integer N2 such that

cвЂ˜

I?. Y,,l <

n>N˜z j

3(1.x1 + 1)

DonвЂ˜t lose sight of the fact that since E and 1x1 arc fixed real numbers, so is

e/[3(lx + l)]. The 3s in these expressions come from the fact that there are three

terms in the each of the cxprcssions above. In order to make this expression less

then cвЂ˜; WC will make each of the three terms less than z/3. To make the first

te˜tn; Irlly y,,l. less than 43, we want Iy y,,/ < ˜/(31x1). We add an extra

1 to the denominator on the right to handle the case where x might bc I). Take

N = max{N,, NZ}. Then. if n 2 N.

Here, we have used the facts that

L<l IYI -1 ˜1

вЂ™ lyl + 1 < l,

1x1 + 1 1x1 + 1 < l, IyI + 1 < l,

and &<l ==a 0= -cl

3

Another important property of sequences is that they preserve weak order relations,

as the following theorem shows.

Let (x,}_˜ be a convergent sequence with limit x, and let b

Theorem 12.4

be a number such that x, 5 b for all n. Then, x 5 b. If n, 2 b for all n, then

x 2 b.

Proof We will prove only the first statement. The proof of the second statement

1is almost identical. Suppose, then, that x. 5 b for all n, and suppose that x > b.

Choose E so that 0 < F < x b; then, b < n 6 and I,(x) = (x - E, x + s)

lies to the right of b on the number line. There is an integer N such that for

all n 2 N, x. E I&). For these x.вЂ˜s, b < xвЂќ; this is a contradiction to the

hypothesis that all the x,,вЂ˜s were C b. W

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