This set of free lecture notes by Fields Medal winner Vaughan Jones are excellent. In that they are virtually verbatim, you get the full presentation as if he was speaking directly to you rather than a summary.They are very accessible, and assume no prior knowledge. They build from the beginning adding insight along the way.
As one of the great mathematicians, he uses his insight in presenting his own treatment of the material and makes his train of thought very clear.
This would be my first choice covering what you ask for and more.
For additional consideration: Pugh's "Real Mathematical Analysis" (as mentioned above) is also excellent, likewise with examples and pictures which are quit beneficial.
Here is a short review from Promys:
An absolutely fantastic introduction to Analysis, it has excellent exposition and is full of great examples and over 500 (good) exercises. A counterpart to Rudin, Pugh always builds up machinery first and uses it to provide very clear proofs that grant a good sense of "why" something is true. Judging by page count and the amount of material it covers, it seems that it must be as concise as Rudin, yet it reads very easily.