Homework assignments for M361K
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Nr Section : Problems
-----------------------------------------
[1]    ... : read sections 1 and 2 in the notes posted here
       1.1 : 7,21
       1.2 : 2
       ... : A1     (see below)
       1.3 : 6,13
----------------------------------------- due Fri Jul 19
[2]    2.2 : 4,12
       2.3 : 5,11
       2.4 : 2
       ... : B1     (see below)
----------------------------------------- due Tue Jul 23
[3]    2.4 : 7,8
      11.1 : 6
       ... : A2,B2  (see below)
----------------------------------------- due Fri Jul 26
[4]    3.1 : 1abd,3abd,5
       ... : A3,B3  (see below)
----------------------------------------- due Tue Jul 30
[5]    3.1 : 12,16
       3.2 : 6c,14a,18,22
       ... : B4     (see below)
----------------------------------------- due Fri Aug 2
[6]    3.3 : 4
       3.4 : 7d     (see Remark R1 below)
       ... : C1,C2  (see below)
       3.5 : 2
----------------------------------------- due Tue Aug 6
[7]    ... : C3,B5,A4  (see below)
       4,1 : 10,12ac
----------------------------------------- due Fri Aug 9
[8]    ... : B6     (see below)
       4.2 : 3
       5.1 : 5
       5.2 : 10
       5.3 : 1,4
----------------------------------------- due Tue Aug 6
[9]    ... : D1,D2   (see below)
       5.4 : 2
       6.1 : 1d,2
       ... : D3      (see below)
       6.2 : 7,13
       7.1 : 2,6,8
----------------------------------------- just practice problems


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NOTE: NO LATE HOMEWORK

Unless stated otherwise in class,
homework is assigned every Tuesday and Friday,
and collected in class
the following Friday and Tueday, respectively.
Late homework will not be accepted.
But as announced, the lowest homework score
gets dropped before computing the homework average.

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REMARKS

R1. For Problem 3.4.7 you can use the following

Fact. The sequence of real numbers xn=(1+1/n)n converges.
The limit is known as Euler's constant and is denoted by e.
Its value is e=2.71828...
(See also Example 3.3.6 in the book.)

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PROBLEMS A1 .. A4 

A1. Prove that every non-empty finite subset
of a totally ordered set contains a maximal element.
Hint: The Principle of Mathematical Induction is needed for this.

A2. Let (S,d) be a metric space.
Let u and v be two distinct points in S.
Show that there exists an open ball U centered at u,
and an open ball V centered at v,
such that the intersection U∩V is empty.

A3. Let (S,d) be a metric space, let A be a closed subset of S,
and let (x1,x2,...) be a sequence of points in A.
If this sequence converges to a limit x∈S, prove that x∈A.

A4. Consider the set P∗ = P ∪ {∞},
where P is the set of all positive integers and ∞ is some extra element.
For m,n ∈ P define d∗(m,n)=|1/m - 1/n| and d∗(m,∞)=d∗(∞,m)=1/m and d∗(∞,∞)=0.
Show that d∗ is a metric on the set P∗.
Considering the metric space (P∗,d∗),
show that the sequence (1,2,3, ...) has a limit, namely ∞,
and that ∞ is the only limit point of P∗.

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PROBLEMS B1 .. B6 

In what follows, R denotes the set of all real numbers.
And to simplify notation, we will identify R2 with RxR.

By definition, an interval is a set S of real numbers
with the property that if x∈S and z∈S and x<y<z, then y∈S.

B1. Prove that the following sets, where a<b, are intervals.
Prove that every non-empty interval is of one of these types.

[a,a]={a},
[a,b]={x∈R: a≤x≤b},  [a,b)={x∈R: a≤x<b},
(a,b]={x∈R: a<x≤b},  (a,b)={x∈R: a<x<b},
(-∞,+∞)=R,
(-∞,b]={x∈R: x≤b},  (-∞,b)={x∈R: x<b},
[a,+∞)={x∈R: a≤x},  (a,+∞)={x∈R: a<x}.

B2. Define d: R2xR2→R by setting d(x,y)=|x0-y0|+|x1-y1|
for any two points x=(x0,x1) and y=(y0,y1) in R2.
Show that d is a metric on R2, and that
the disk S={x∈R2: x02+x12<1} is open for this metric.

B3. Use the definition of the limit of a sequence
to show that the following sequences have no limit:
X=(0,1,0,1,0,1, ...), Y=(0,1,2,3,4,5, ...).

B4. Let X=(x0,x1,x2,...) be a sequence of real numbers
in an interval [a,b] with a>0,
and let x be a real number in the same interval.
Let k be an integer larger than 1.
Prove that xnk→xk if and only if xn→x.

Hint: Recall that (xnk-xk)=Cn(xn-x), where
Cn=xnk-1x0+xnk-2x1+xnk-3x2+...+xn2xk-3+xn1xk-2+xn0xk-1.
Prove first that there are positive real numbers
A and B such that A≤Cn≤B for all n.

B5. Let f(x)=x3-x1/3 for all x in [a0,b0]=[½,2].
For n=0,1,2,...
let xn be the midpoint of [an,bn], and set [an+1,bn+1]
equal to [an,xn] if f(xn)≥0, or equal to [xn,bn] if f(xn)<0.
Show that the sequence (x0,x1,x2,...)
converges to some x∈[a0,b0], and that f(x)=0.

B6. Use the Contraction Mapping Theorem to show that the
equation 3-2/x3=x has a unique solution in the interval [2,3].

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PROBLEMS C1 .. C3

Notation. Given a sequence of real numbers X=(xm,xm+1,xm+2,...),
the sequence of partial sums S=(sm,sm+1,sm+2,...), defined by
sn=xm+xm+1+xm+2+...+xn for n≥m is called the series associated with X.
Instead of S one also writes Σn≥mxn.
In sloppy (but common) notation, the same symbol Σn≥mxn
is also used for the limit of S, if the series converges.

C1. Let 0≤r<1. The geometric series with ratio r is the
sequence S=(s0,s1,s2,...) defined by sn=1+r+r2+...+rn.
(a) Show that (1-r)sn=1-rn+1 for all n.
(b) Prove that the series converges.
(c) Use (a) and (b) to prove that the series converges to s=1/(1-r).

C2. Let 0≤r<1. Let X=(x0,x1,x2,...) be a sequence of
positive real numbers satisfying xn+1≤rxn for all n.
Set A=x0 and B=A/(1-r).
(a) Show that xn≤Arn for all n.
(b) Prove that xn+xn+1+...+xm≤Brn for all n and all m>n.

C3 (telescoping series). Let T=(t1,t2,t3,...) be any
sequence that converges to 0. Prove that Σn≥1(tn-tn+1)=t1.
Use this to show that Σn≥11/[n(n+1)]=1.

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PROBLEMS D1 .. D3

Theorem. The limit Exp(x)=lim(1+x/n)n exists for every real number x.
The resulting function Exp has the following properties:
(a) Exp(x+y)=Exp(x)Exp(y) for all real numbers x and y.
(b) |Exp(x)-1-x|≤x2 whenever |x|≤ 1/2.

Note. By (b) we have Exp(0)=1,
and setting y=-x in (a) one finds Exp(-x)=1/Exp(x).
With a bit more "algebra" one also finds that
Exp(q)=eq for every rational number q,
where e=2.71828... is Euler's constant defined earlier.

D1. Using the Theorem above, prove that
(a) Exp is continuous at zero.
(b) Exp is continuous at every real number c.
    Hint: Exp(x)-Exp(c)=Exp(c)[Exp(x-c)-1].

D2. Using the Theorem above, prove that
(a) Exp(x) is positive for all x.
(b) Exp is strictly increasing and unbounded above.
(c) Exp is one-to-one and maps R onto (0,+∞).

Note. This implies that Exp:R→(0,+∞) has an inverse Exp-1:(0,+∞)→R.
The inverse of Exp is also denoted by Ln (the natural logarithm).
It can be used e.g. to define ax as Exp(x Ln(a)) for every a>0.

D3. Using the Theorem above, prove that
(a) Exp is differentiable at zero, and Exp'(0)=1.
(b) Exp is differentiable at every real number c, and Exp'(c)=Exp(c).
    Hint: Exp(x)-Exp(c)=Exp(c)[Exp(x-c)-1].