[wish5031]

As others have mentioned, Tao has fantastic resources, and his blog might be worth looking at. Rudin is a classic. I studied real analysis through “Advanced Calculus” by Fitzpatrick which has some great material.

For more advanced analysis (esp. functional analysis) I would look at Kreyszig or Hunter & Nachtergaele.

The best way to prepare imo is to just do proofs between now and the start of the course. Try to find practice proof problems online and see if you can do them or find an entry-level book on discrete math. Problem-Solving Strategies by Engel is a good but slightly more advanced book for a beginner.