Unformatted text preview: Problem 4.9. Show that lim J5 = 1 (and conclude ﬁam this that ﬁrmly; z 1
for each a > 0). Solution. Note that W = ("L/F )2. An easy inductive argument shows that ﬁfﬁ > 1 holds for each n. Thus, we can write "7J5 = liJt,l with In > 0.
Since (1 «l a)" 2 1 + no holds for each n and each a a 0 (see Problem 2.13).
we get ﬁ=(m)"=(1+xn)"zl+nx,. and so a < x" m % — i. This implies £1me = 0. Therefore. W=.(~/"~/r7)1=a+xn)1—u. An altemale proof goes as follows: By L’Hépital’s Rule, we have link...” l? = 0, and so Emu...” 15,} = 0. Therefore. using that the exponential function is
continuous, we infer that lim W: lim chi"! =eo=L
ﬂﬁm n—om
For the parenthetical pan, assume ﬁrst a > I. Then it is easy to see that 1 5
{73 5 45 holds true for all n > .2. Consequently. by the “Sandwich Theorem,”
weswdmtlimc/E: 1. IfO < a < Lthené > 1,andsolimo/l_= rim5}»; = 1, a
from which it follows that lim {/5 a 1 holds true in this case. too. Problem 4.10. If [an is a sequence of strictly positive real numbers, then show
that . . X . . . . I +1
hmmf "+1 5i:m1nf.:/x,, 5 hm sup Mn. 5 hm sup "—~.
"“W In ""W orgon nwn In Conclude from this that if lirn ’21—? exists in R, then Hindi; also exists and
lim‘ﬂ': = lim "—;—:‘1. Solution. Let {x,.} be a sequence of real numbers such that x, > 0 holds for
each n. We shall establish lim sup {/ﬁ' 5 limsup ﬁ—“fl and leave the similar proof
of the other inequality for the reader. Put . xn+l m m xitH
x=ltmsup =/\V—,
L, 11'
n=1 I:=n and note that if .r = 00. then there is nothing to prove. So,wecanassume x < 00.
Let 5 > 0 be ﬁxed. Then there exists some I: such that ‘2' < .r + 8 holds
for all r: a 1:. Now, for n > k we have In = _rn_.£a:_l on an 5(x+£)"kx, =(x+g)"c,
In—l Jim—z a} where c = 1,1,: + c)‘* is a constant. Therefore. {/5 5 (x + ark/E holds for
each n 2 It and so. in View of lim (YE = I (see Problem 4.9) and Problem 4.8,
we infer that limsupﬂ 5 limsup(x + SN/E = {at +3) lim #5 = a: + e.
n—roo nwoo ""‘m Since 2 >9 is arbitrary, we inferthal limsupjﬂs x = limsop‘"' x.‘ ...
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 Spring '10
 GuoliangWu
 Math

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