Week 6: So many proofs

This week we learned proofs for non-boolean functions such as floor and ceiling functions, and for limits, which I found a little challenging. (I am also deeply distraught that proof by illegibility is not valid. Fine... you caught me.)

If I had thought that the definition of a limit was confusing, the proof for it is something else. From last week, we learned that






can be defined as




so that as ε gets smaller, x approaches 2, and as δ gets smaller, f(x) = x² approaches 4. To prove this, we must show that for all positive values of ε, there is a positive value δ such that if | x - 2 | < δ then | x² - 4 | < ε. The key to this proof is to choose an appropriate δ, in terms of ε, such that allows us to transition from | x - 2 | < δ to  | x² - 4 | < ε. This is a complicated step, and involves some clever manipulations. It took several minutes of stupefied staring at the lecture notes to fully understand the choice of δ for this proof. Further stupefied staring at the mass of confusion that is course notes and I begin to see that writing proofs is not just about thinking logically; it involves searching outside the box for a creative solution, then presenting it logically.