There are horde of good books in all fields of mathematic. What you need is something you can learn from, not only the best and most glorious of this books.
Books with so much problems and exercises with their hints and solutions are very appetizing. But what you really need is a mature and deep grasping of basics and concepts. After all thats all what you need to tackle this exercises with even a surprising ease and fun.
Analysis is among the most reachable field in math after high school, and a fare knowledge is required in most of the other fields for beginners.
I do understand the emphasize on solutions. I do because we all deal with self study, at least sometimes, and solutions and hints are crucial to make an evaluation of your own work.
If you are really serious you will soon find out that what you really need are hints not solutions.
Needless to say hints or solutions are supposed to be a last resort , when there seems to be no way out. Even then a hint is better taken only partially. And by the way : when tackling problems,It is when there seems be NO WAY OUT that the actual LEARNING process takes place.
I encourage you to take a deep look into The Trillia Groupe funded,and fee, Zakon's books: Mathematical Analysis I which followed by another volume, but to get some basics ,Basic Concepts of Mathematics might be a good place to start.
In the third mentioned book , this was mentioned:
Several years’ class testing led the author to these conclusions:
1- The earlier such a course is given, the more time is gained in the
follow- up courses, be it algebra, analysis or geometry. The longer
students are taught “vague analysis”, the harder it becomes to get
them used to rigorous proofs and formulations and the harder it is
for them to get rid of the misconception that mathematics is just
memorizing and manipulating some formulas.
2- When teaching the course to freshmen, it is advisable to start with
Sec- tions 1–7 of Chapter 2, then pass to Chapter 3, leaving Chapter
1 and Sections 8–10 of Chapter 2 for the end. The students should be
urged to preread the material to be taught next. (Freshmen must learn
to read mathematics by rereading what initially seems “foggy” to
them.) The teacher then may confine himself to a brief summary, and
devote most of his time to solving as many problems (similar to those
assigned ) as possible. This is absolutely necessary.
3-An early and constant use of logical quantifiers (even in the text)
is ex- tremely useful. Quantifiers are there to stay in mathematics.
4- Motivations are necessary and good, provided they are brief and do
not use terms that are not yet clear to students.
In the second book , This was mentioned :
Several years’ class testing led us to the following conclusions:
1- Volume I can be (and was) taught even to sophomores, though they only
gradually learn to read and state rigorous arguments. A sophomore
often does not even know how to start a proof. The main stumbling
block remains the ε, δ-procedure. As a remedy, we provide most
exercises with explicit hints, sometimes with almost complete
solutions, leaving only tiny “whys” to be answered.
2- Motivations are good if they are brief and avoid terms not yet known. Diagrams
are good if they are simple and appeal to intuition.
3- Flexibility is a must. One must adapt the course to the level of
the class. “Starred” sections are best deferred. (Continuity is not
4-“Colloquial” language fails here. We try to keep the
exposition rigorous and increasingly concise, but readable.
5- It is
advisable to make the students preread each topic and prepare ques-
tions in advance, to be answered in the context of the next lecture.
6- Some topological ideas (such as compactness in terms of open
coverings) are hard on the students. Trial and error led us to
emphasize the se- quential approach instead (Chapter 4, §6).
“Coverings” are treated in Chapter 4, §7 (“starred”).
7- To students
unfamiliar with elements of set theory we recommend our Basic
Concepts of Mathematics for supplementary reading. (At Windsor, this
text was used for a preparatory first-year one-semester course.) The
first two chapters and the first ten sections of Chapter 3 of the
present text are actually summaries of the corresponding topics of
the author’s Basic Concepts of Mathematics, to which we also relegate
such topics as the construction of the real number system, etc.
I did not take this points very seriously, until i started reading and working on it.
It is hard to find yourself completely stuck somewhere: It seams that all have been packed for a person who is learning on his own. Hints are provided anywhere whenever needed. In many occasions there are questions like "...Why?" which helps in following the text rigorously.