So far I wasn't able to find one single open source set of lecture notes or textbook with $\LaTeX$ source code available on "introductory real analysis" that could suit my needs. It seems to me that you can find either sloppy calculus books or books whose very first chapter is on Banach Spaces. Could you point out something in the middle: that is, some open source lecture notes that cover the same material covered by Rudin and Ross and treat the subject with rigour?
(Migrated from the comments.)
Below are two different textbooks that may fit the bill.
First, note that the OP has previously asked a more general question about textbooks here on MSE. That earlier question included an answer with the Intro to Real Analysis book here, which has the LaTeX on the same page: scroll down to Source. The textbook is available as a free pdf, too.
The text has been used at a bunch of quality institutions, from UW-Madison to UC-Berkeley, and it has a favorable MAA review from 2013 (of the December 2012 version of the text). Although it has been updated as recently as December 2014, the citation on google scholar is given as:
Lebl, J. (2011). Basic analysis-introduction to real analysis. AMC, 10, 12.
For another free text (with available TeX) see here for Trench's (December 2013) work.
The recommended citation there is given as:
Trench, William F., "Introduction to Real Analysis" (2013). Books and Monographs. Book 7.
Using a clear and informal approach, this book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.
Moreover, the page notes that:
This book was previously published by Pearson Education. This free edition is made available in the hope that it will be useful as a textbook or reference.
A citation for the earlier version is:
Trench, W. F. (2003). Introduction to real analysis. Upper Saddle River, NJ: Prentice Hall/Pearson Education.