Homework assignments for M361K
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Nr Section : Problems
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[1]    1.1 : 7,21
       1.2 : 2,8
       1.3 : 6,13
---------------------------------------- due Fri Jul 13
[2]    2.2 : 4,12
       2.3 : 5,6,11
       2.4 : 2,7
---------------------------------------- due Tue Jul 17
[3]    3.1 : 5,9,12,16
       ... : A1,A2      (see below)
---------------------------------------- due Fri Jul 20
[4]    3.2 : 6c,14a,17,18,22
       ... : A3         (see below)
---------------------------------------- due Tue Jul 24
[5]    3.3 : 4
       3.4 : 7d    (see Remark R1 below)
       3.5 : 2b,13
       ... : B1         (see below)
---------------------------------------- due Fri Jul 27
[6]    4.1 : 10b
       4.2 : 1c,2b,3
       5.1 : 5,10,12
       5.2 : 1b
---------------------------------------- due Tue Jul 31
[7]    5.3 : 1,4
       5.4 : 2
       ... : C1         (see below)
       6.1 : 1d
---------------------------------------- due Fri Aug 3
[8]   6.1  : 2,8c,9,14
      6.2  : 7,14,15
---------------------------------------- due Tue Aug 7
[9]    7.1 : 2d,6,8
       7.2 : 8,12,16
       7.3 : 4,6
---------------------------------------- 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...

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PROBLEMS A1 .. A3 

A1. Let a and δ be fixed but arbitrary positive real numbers.
Show that there exists a positive real number ε
such that |b-a|< ε implies |b2-a2|< δ.

A2. 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.

A3. Let 0<r<1.
Let (d1,d2,...) be a sequence of nonnegative real numbers, satifying
dn+1 ≤ rdn for all n≥1. Show that dn ≤ d1rn-1 for all n≥1.
Given a fixed but arbitrary m≥1,
let (D1,D2,...) be a sequence of nonnegative real numbers, satisfying
Dk ≤ dm+dm+1+ ... +dm+k-1 for all k≥1.
Show that Dk ≤ dm(1+r+r2+ ... +rk-1) for all k≥1.
Then multiply by 1-r to prove that (1-r)Dk ≤ dm for all k≥1.

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PROBLEMS B1 .. 

B1. 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 cluster point in this space.

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PROBLEMS C1 .. 

C1. Let f be a continuous real-valued function on an interval I=[a,b].
Show that for every ε>0 there exists a piecewise constant function h on I,
such that |f(x)-h(x)|<ε for all x∈I.

Note: A function h on I is said to be piecewise constant if the range of h,
as well as the set of points in I where h is discontinuous,
are both finite sets.
(So the graph of h consists of finitely many horizontal line segments.)