1549901369-Elements_of_Real_Analysis__Denlinger_

82 Chapter 2 • Sequences

18. Prove that if all terms of a convergent sequence { Xn} lie in a closed interval
    [a, b], then its limit is also in [a, b]. What if [a, b] is changed to (a, b)?


19. Let e E JR. be fixed. Prove that Vp E N , if lei < 1, lim nPen = 0.
    n->oo


20. '<Ix E JR., let [x] = the greatest integer less than or equal to x, and let
    e E R Prove that
    . [en]
    (a) hm - = e.
    n-+oo n
    (b) If e > 0, lim [~] n = 0.
    n--+oo en

2.4 Divergence to Infinity

Definition 2.4.1 Suppose {xn} is a sequence of real numbers. We say that

(a) {xn} diverges to +oo ( lim Xn = +oo) if

n->oo

VM > 0,3no EN 3 n 2: no==:> Xn > M;

(b) {xn} diverges to -oo ( lim Xn = -oo) if

n->oo

VM > 0, 3no EN 3 n 2: no==:> Xn < -M.

Note that this definition implies that if lim Xn = +oo (or -oo) then {xn}
n->oo
is unbounded, hence {xn} cannot converge (why?). So we will not say that {xn}
converges to +oo or -oo, or that lim Xn exists in these cases, even though we
n->oo
use the notation lim Xn.
n->oo

Example 2.4.2 Consider the limit statement lim (
3
n

2

2

n) = +oo.

· n->oo 5n + 23

3n^2 - 2n

(a) After how many terms are we guaranteed that > 100?

5n+ 23

(b) For arbitrary M > 0, after how many terms are we guaranteed that

3n^2 - 2n M?

--->.

5n+ 23

. 3n^2 - 2n
Solution: (a) We want an no EN 3 n > n 0 ==:> > 100. Now,

• 5n+ 23
  3n^2 - 2n 3n^2 - 2n 'f
  ---> ,1n> 24
  5n + 23 5n + n -

3n^2 - 2n

6

n , if n 2: 24

3n-2

• 6
  
  
  • , if n 2: 24.