2.2 Algebra of Limits 69Case 2 (L > 0): Since Xn---+ L , 3 n2 EN 3 n :'.'.'. n2::::} lxn - LI < c.JL. Let
no= max{n1,n2}. Then
n :'.'.'. no ::::} IFn - ./Li = lxn -LI < lxn - LI < c.JL = €. That is,
Fn + .JI, .JI, .JI,
n :'.'.'.no::::} IFn -./Li < c. Therefore, y'Xr;"---+ .JL. •Example 2.2.14 Use the "Algebra of Limits" to prove lim (
2
n +
3
) = ~-
n-+oo 3n - 7 3S 1 · · 1-1 iM(^2) n + (^3) - (^2) + ~ Th b Th d 7
o ution. vn E n, -
3
--- -- 7. us, y eorems 2.2. 13 an 2.2.6,
n-7 3-:;:;:
r ( 2n + 3) r ( 2 + ~ ) nl!..1! ( 2 + ~)
n~1! 3n - 7 = n~1! 3 - l = -1-im-~( 3 --~1~)
n n--+oo n
2 + 3 nl!..1! ~ 2 + 0 2
3 - 7 lim .1 3 - 0 3
n--+oo n
lim 2 + lim ~
n--+oo n--+oo n
lim 3 - lim l
n-+oo n--+oo n
D
ONLY THE "TAIL" MATTERS!
Definition 2.2.15 For a fixed m E N, the m-tail of a sequence {xn} is the
sequence
Tm= {xm,Xm+1,·· · ,Xm+n,· ··}
= {xk}~m
= {xm+n};;_o=O
That is , them-tail of { Xn} is the sequence that results when the first m -1
terms of { Xn} are deleted.
Theorem 2.2.16 A sequence {xn} converges to L <=? all of its m-tails Tm
converge to L <=? one of its m -tails Tm converges to L. (That is, for fixed
m EN, lim Xn = lim Xm+n·)
n--+oo n--+oo "-.
Proof. Exercise 18. • ' ,.,,.,,.. >
In this sense, only the "tail" of a sequence matters in considerations of
convergence and limit. The first few (finite number) of terms of a sequence do
not matter.
- To be strictly correct, the equations b elow must b e read from right to left, to avoid writing
limits before we a re sure that they exist.