This chapter turned the whole course into a single definition and then took it apart. The questions here arrive without saying which idea they test, which is the state in which mathematics is normally met.

Every tool in the chapter is now on the bench at once. Choosing which one a question calls for is the part that carries over to everything that follows.

A sequence converges when every tolerance is eventually met. Denying that flips all three quantifiers. Limits are unique, convergent sequences are bounded, limits survive arithmetic, and a monotone bounded sequence converges to a supremum it need never reach.

• The order of the quantifiers is the definition; $N$ may depend on $\varepsilon$.
• Only finitely many terms may sit outside a tolerance, so any finite prefix is irrelevant.
• Bounded does not imply convergent, but convergent does imply bounded.
• Splitting a tolerance into shares that add back to it is the pattern behind every proof here.
• Bolzano–Weierstrass: every bounded sequence has a convergent subsequence.
• The Cauchy criterion: a sequence of reals converges exactly when its terms eventually crowd together.
• A bounded sequence converges exactly when its upper and lower limits agree.

Great work! You now know how to:

• Choose a place for a given tolerance without hesitating
• Name the right theorem for a question about a sequence
• Use the squeeze theorem, Bolzano–Weierstrass and the Cauchy criterion where each belongs