Basic Analysis

This page lists the definitions and theorems from basic real and general analysis that you’ll need to follow the posts here.  I’m not going to go into a lot of details about these ideas, but will give some basic definitions and statements of theorems, then link you to corresponding pages on Wikipedia to let you read some more if you’re interested.

Supremum and Infimum: We say that a set of real numbers, , is bounded if there exists a such that for each we have .  More generally, we say that is an upper bound of if for each .  Similarly, is a lower bound of if for each .  Simply saying that is bounded means that is both bounded above (has an upper bound) and bounded below (has a lower bound).  The supremum of is the least upper bound of and is denoted .  Obviously, a supremum will only exist if a set is bounded above, but that alone is not a sufficient condition for the existence of the supremum.  For example, consider the all of the rational numbers that are strictly less than : .  In this set is bounded above, but no supremum exists:  For any rational number that you find, you can always find another one that’s closer.  In the space , however, there is a least upper bound, namely .  A space is said to have the completeness property if every set that is bounded above has a supremum.

The infimum is defined similarly to the supremum: the infimum of a set is the greatest lower bound of .  Naturally we require that be bounded below for an infimum to exist, but just like with supremum, this isn’t enough.  It’s not difficult to show that if a set has the completeness property (so every set bounded above has a supremum), then every set that is bounded below has an infimum (just take the supremum of the set of all lower bounds).

Sequences: A sequence is simply a function whose domain is the natural numbers.  In the case of real-valued sequences, a sequence is a function .  For complex-valued sequences, it’s a function .  In some arbitrary space , a sequence in is a function .  Basically, a sequence is just an ordered collection of elements of the space where elements can repeat (so it’s not simply just a subset of elements).

Countable and Uncountable: I’m just going to go ahead and link you to Wikipedia: Countably Infinite, Uncountable.

Bolzano-Weierstrass Theorem: Bolzano-Weierstrass states that every bounded real-valued sequence has a convergent subsequence.  The proof of this has two not-too-difficult parts: First you show that every monotonic sequence converges if and only if it is bounded, then you show that every bounded sequence has a monotonic subsequence.  (The actual statement of this theorem on Wikipedia is more general, but this is what I’m referring to if I say “Bolzano-Weierstrass.”)