I've spoken with more than one person who made it through Real Analysis intact by reading through "
Introductory Real Analysis" by Kolmogorov and Fomin. There's a Dover version that you can probably find for $12 used... It was where I started, but I know several people who found it invaluable after struggling with other texts.
Introductory Real Analysis (Dover Books on Mathematics) https://www.amazon.com/dp/0486612260/ref=cm_sw_r_cp_apa_a62T...
Come to think of it, there are a lot of good Dover books on mathematics.