Let (xn) be a sequence with positive terms. (a) Prove
the following: lim inf xn+1/ xn ≤ lim inf
n√ xn ≤ lim sup n√xn ≤
lim sup xn+1/ xn .
(b) Give example of (xn) where all above inequalities
are strict. Hint; you may consider the following sequence
xn = 2n if n even and xn = 1 if n
odd.
Let (xn), (yn) be bounded sequences.
a) Prove that lim inf xn + lim inf yn ≤
lim inf(xn + yn) ≤ lim sup(xn +
yn) ≤ lim sup xn + lim sup yn.
Give example where all inequalities are strict.
b)Let (zn) be the sequence defined recursively by
z1 = z2 = 1, zn+2 = √
zn+1 + √ zn, n = 1, 2, . . . . Prove that
(zn) is convergent and find its limit. Hint; argue...
Let a sequence {xn} from n=1 to infinity satisfy
x_(n+2)=sqrt(x_(n+1) *xn) for n=1,2 ......
1. Prove that a<=xn<=b for all n>=1
2. Show |x_(n+1) - xn| <= sqrt(b)/(sqrt(a)+sqrt(b)) * |xn -
x_(n-1)| for n=2,3,.....
3. Prove {xn} is a cauchy sequence and hence is convergent
Please show full working for 1,2 and 3.
4. (Reflected random walk) Let {Xn|n ≥ 0} be as in Q6. Show that
Xn+1 = X0 + Zn+1 − Xn m=0 min{0, Xm + Vm+1 − Um+1}, where Zn = Xn
m=1 (Vm − Um), n ≥ 1. Q5. (Extreme value process) Let {In|n ≥ 0} be
an i.i.d. sequence of Z-valued random variables such that P{I1 = k}
= pk, k ∈ Z and pk > 0 for some k > 0. Define Xn = max{I1,
I2, ·...
Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m >
1, m ∈ N such that |sn − m| = 1/3 (this says that every term of the
sequence is an integer plus or minus 1/3 ). Show that the sequence
sn is eventually constant, i.e. after a point all terms of the
sequence are the same