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I'm asking because I plan on taking Real Analysis later and I'd like to gain a better understanding of Calculus.

I have read "Trig on Tears" and skimmed through a couple Algebra and Trigonometry textbooks, I know the basic Logic, and I've followed through perfectly on Prof. Leonard's Calculus I playlist. Additionally, I've done several practice sets involving related rates, extremely basic differential equations, calculus with inverse trig functions, integration by parts, some introduction to the delta-epsilon definition of limits (and proving limits from this), linear approximations, optimizations, and some additional techniques (like sign analysis, concave up/down, increasing/decreasing, curve sketching).

If I continue to read this text, the next chapters involve Relations, Functions, Proof by Mathematical Induction, and something about series/sequences, in that order. Just thought I'd include what I'd be missing if I stop after this chapter. I don't have enough time to continue once this next term starts, so I may have to hold off until summer for the rest.

I'm not sure if proof by induction will be necessary for Spivak's Calculus, but I will be later taking Discrete Mathematics, anyway.

Or should I trust that my professor will cover enough material and I can wait a couple years to get into actual proof writing?