No clear-cut best option for Introductory Real Analysis course

[bradleygrant]

BLUF: I've been racking my brain trying to find a 1) good 2) free 3) introductory course in Real Analysis 4) with video lectures, 5) assignments and 6) an easily accessible book 7) that isn't Rudin. I want this to go in the OSSU Core Mathematics slot for Intro To Analysis.

I need some help.

Details: To clarify:

1) good = acceptably high quality for inclusion in OSSU. 2) free = acceptably low price for inclusion in OSSU; freemium courses considered. 3) introductory course = suitable for a beginner to self-study. No senior-level/grad work, no starting assumption of topology. 4) with video lectures = Watching somebody write proofs on the board -- preferably that match the topic, assignments and book! 5) assignments = a selection of homework practice, preferably with worked proofs and solutions. 6) an easily accessible book = one that's open-source, or that's still in print and readily available for free or a reasonable price. 7) that isn't Rudin = it's a high-quality text for the junior/senior major level, but painfully unapproachable if you're still learning how to work proofs. MIT finds it too difficult for a first book for MIT math majors with instructor and TA support. How are we going to recommend first-timers self-study it??

Here's what I've found so far, loosely ordered from Most to Least Complete:

Introduction to Real Analysis at Bethel University -- has 1, 2, 3, 4, 5, 7.

This is taught by Professor Bill Kinney (https://infinityisreallybig.com) at Bethel University specifically as a first course for majors. It is a gentle introduction to the proof and he states that this course alone may not be enough preparation for grad school, indicating it's not a Real Analysis 1 course for upperclassmen. Additional course materials were recently made available by the instructor at https://drive.google.com/drive/folders/1QtcZHip4x8_hxc1gACgWsHDfdngo36Vj?usp=sharing. By all respects it looks like a great option, but the book is Gordon 2e which runs $120 and there aren't many used ones in circulation.

MIT 18.100A -- has 1, 2, 3, kinda 5, 6, 7.

Mattuck is an excellent book available for very cheap. Assumes only Calculus 1. No video lectures and no solutions mean it's hard to teach off of.

Real Analysis 1 at Harvey Mudd -- has 1, 2, 4, 5, 6.

The analysis course currently linked in the OSSU Math curriculum. Honestly, Prof. Su seems to support his class well and may make Rudin palatable for the first-timer, but I'm skeptical.

Several courses assigning homework out of Abbott's text, often regarded as a good first choice text, but without videos or solutions

I'd like to find some video lectures of a competent professor teaching out of Abbott, that would be about perfect for this course.

Adapt Abbott to the Bill Kinney Bethel lectures and cobble a complete course together

A concept I had. Abbott appears to follow this course pretty well, read chapters 1, 2, 3, 4, 5, 7, 6, 8. Can't guarantee coverage of topics but it seems to be fairly close by.

Just teach yourself out of Book Of Proof and Mattuck/Abbott

The current state of things, but I think we can do better.

I checked EdX and Coursera, and they seem to be fresh out of real analysis.

So, a few things I'd like us to consider:

• How does OSSU feel about recommending a course with a required textbook that isn't readily available? The library is an option
• Am I even focusing on the right ideals?
• Am I correct in assuming a course in baby Rudin is "too advanced" for a first analysis class (for mere mortals who didn't get into Harvey Mudd)? Does Prof. Su redeem the difficulty?
• Has anybody taken any of these or others, and has other courses to recommend?
• Is there a good course in Advanced/Honors Calculus/Intro Analysis that's using Spivak's Calculus text? I haven't investigated this. Maybe that presents a better option. (Feel free to slap me if that's definitely not a better option.)
• What other resources are out there that I haven't seen yet? Links, links, links, links! :)

And finally, the big takeaway:

I think from what I've seen so far, the Intro to Real Analysis at Bethel is the closest thing we have to the right first analysis course. In order for me to feel good about recommending it, we need to host the documents somewhere that isn't a Google Drive. I don't really know the best way to go about doing that (except perhaps by making a Github Pages site for the course).

[uRSTEnzY]

Here's a very good introduction to Real Analysis textbook that can potentially be used: https://digitalcommons.trinity.edu/mono/7/. There are no video lectures directly associated with the text, but looking at the Table of Contents, we should be able to easily find a Youtube playlist to accompany the text.

[bradleygrant]

Here's another Intro Analysis course for consideration:

UCCS Math 3410 Intro To Analysis (has 1, 2, 3, 4, 5, 6, 7), with video lectures here

This one uses Ross's book, which is another gentle introduction to the proof, available on Springer for $45 and via SpringerLink for free through certain institutions. The course has exam study guides with solutions, 16 weeks worth of lectures, and another 16 weeks worth of lectures with a different professor if you don't like the first one's teaching style.

[bgraham89]

@bradleygrant MIT 18.100A now has a 2020 version with videos here.

[waciumawanjohi]

As 18.100A is currently the recommended course, and including lectures is a significant improvement over the same course from the same institution without lectures, I'm going to open a PR to bump the current recommended course to the 2020 version. This is not meant to close this issue, as feedback is important and assignments with solutions are a proxy that we should hope to attain.