Real Analysis – Exercises
— 1. Words of Caution —
In my book of Quantum Mechanics by Sakurai at a given point in the preface it is said something like: “The student who has read the book but can’t do the exercises has learned nothing!”. And that is a true statement if I ever saw a true statement.
How many times I’ve heard people say: “I understand the theory, but I just can’t do the exercises…” This type of feeling is wrong and counterproductive.
Wrong because if the person had indeed understood the theory he/she would be able to solve more exercises than just the trivial ones, and counterproductive because hiding behind that sham will only make the student’s case worse and worse.
The current subject matter always builds from the previous subject matter and this is a snowball of I understand the theory, just can’t do the exercises that keeps on growing.
Instead, if people are grown up enough to understand that they don’t have to understand everything immediately, and most times have to work hard trying to understand what took many years of the best minds around to accomplish, positive results could be achieved.
Alas!, enough with the moral high ground and let’s get started with an integral part of this blog – Exercises!
Every now and then midsubject exercises will appear and after the class notes are finished a batch of solved exams will be presented.
As a final thought I ask the reader not to immediately read my solution of the exercises but try to work through the exercises first and then go on to my solution and compare them both.
Even if you can’t do it by yourself that effort you put it into it at first will help you understand better what I’ve done.
Also bear in mind that the solutions that I post here are in no way unique or the best solutions, and in some instances they may even be wrong (though I hope this won’t happen too often).
— 2. Exercises —
Exercise 2

Exercise 3 The factorial of a natural number can be defined by recursion by the following relationships: and . denotes the binomial coefficient which is the number of ways you can choose elements from an element set.

Exercise 4 Prove by induction .
For it is which is a valid statement. Now we want to prove that the validity follows from the validity of for some .
In conclusion 
Exercise 5 Prove by induction on that
For it is
Now our inductive hypothesis is
and we want to prove
It is

January 4, 2009 at 12:23 am
you lost me where it said 1. Using AI to AV ….good luck!
January 4, 2009 at 12:30 am
Hi wesley. When I said: “Using AI to AV” it was short for: “Using Axiom I to Axiom V”. And in the other bit where I mention TVII it is Theorem VII. But I just didn’t want to type all of that.
Any more comments are more than welcomed so that any other point that wasn’t made explicit enough can be explained.
January 5, 2009 at 12:27 am
hey, can we pause the snow kinda distracted me when i tried to understand that stuff tnx,
for the statement above you should tell what the abbreviation stands for the first time you use it so if some one does not know they could refer to that(you lost me there too lol)
so far no luck getting this stuff ill try some more tomorrow.
January 5, 2009 at 1:35 pm
Ok. The post is updated. No more shady acronyms for now on.
February 5, 2009 at 12:25 pm
[…] was proven in here we can write . Writting out the terms we […]
May 18, 2009 at 5:02 pm
he as got a pc program that makes any thing realated with maths