Well, there is no shortage of very good and VERY SIMPLE and rigorous intros to analysis books out today:

How to Think About Analysis by Lara Alcock.

Understanding Analysis by Stephen Abbot.

Mathematical Analysis and Proof by David Stirling.

Numbers and Functions: Steps into Analysis by Burn.

Analysis: With an Introduction to Proof by Steven Lay.

A First Course in Mathematical Analysis by David Brannan.