Section 5.1

Section 5.1 begins the construction of the real numbers by introducing sequences, and by defining the notions of Cauchy sequence and bounded sequence.

The definition of Cauchy sequence has three nested quantifiers ("for every … there exists … for every …") and may seem like a lot. But this definition is so important that I recommend memorizing it fully (but make sure you also understand it!).

Also, if you have not encountered the symbols \(\forall\) and \(\exists\) yet, \(\forall\) means "for all" and \(\exists\) means "there exists". These symbols are sometimes used as shorthand instead of writing out the equivalent phrases.