<p>I don’t think you need any academic preparation over the summer. You have two terms of pass/fail to remedy any deficiencies. I think you should either find something exciting and non-academic that you actually enjoy doing, or just spend the summer relaxing, hanging out with friends and having fun. There’s a lot to be said for taking some time to do nothing. Also, what would you hope to gain by memorizing proofs? Caltech wants you to be able to write proofs as a way of training you to think, not because there’s some canon of proofs that everyone needs to be able to reproduce by rote.</p>