I have just started studying ' Principles of mathematical analysis' by Rudin. And I have heard that it is hard to complete . I seriously want to study it and I need some suggestions about how should I proceed by self studying it ?
I have studied abstract algebra and have some basic knowledge about real analysis .
This is not a difficult book. There are books which are difficult, such as the two volume book on Trigonometric series by Antoni Zygmund which is not a good book for a beginner in analysis.
Of course, it goes without saying that you should answer lots of questions (preferably challenging and doable questions), but I would also recommend going through the literature line-by-line. This is a great way to learn the material. When I am faced with a mathematical sentence, I break it down to a simple or broader form that an undergraduate would understand. Here is an example I took from the book:
P302:$~$"Suppose $\phi$ is countably additive on a ring $R$"
Possible interpretation: Imagine we have some algebraic structure, ie, some set with at least one operation on it. Imagine these operations generalise the classic arithmetic operations, such as addition and multiplication. Now take subsets of this abstract set. The set is going to have some form of 'size', and countable additivity gives an abstract interpretation of the size of the set via classical tools we are familiar with in the actual world, such as volume and area.
Does that help?