#cauchy sequences

now, this is an arduous task, after all there are uncountably infinitely many of them. how can we come up with a definition that will possibly apply to all of them? well, it is a teeny bit complicated, but i’ve posted a lot of previous definitions that should help us out with this, i’ve been taking baby steps for any math anxious folks <3

if we take sequences of rational numbers, many of these sequences may converge to another rational number. however, some sequences may want to converge to something else, but because their existence is limited to being rational, they cannot converge to a real number, even though they really want to. this is how the real numbers are defined.

if we take a sequence of rational numbers, it may get pretty complicated, it may bounce back and forth between two rational numbers, and it turns out there is a non-rational number in between those two. if after a while the terms in the sequence are infinitely close to this other number that isn’t rational, then we are able to define it using this sequence, and we call it a real number. weird.