Real Analysis/List of Theorems

Below are a list of all the theorems that are covered by this wikibook. The format for each of them will not be like the theorems found throughout this wikibook however, instead they will be written as a strict if-then statement, without any given statements or explanations. Although this makes each theorem considerably shorter and easier to fit onto one page than by simply copy-pasting each proof, you will not gain the benefit of knowing how the proof is formulated nor the context for most of these theorems (which might be bad when you have no idea what each variable represents).

Note the following:
 * 1) The theorems here does not explicitly define any words — look for the adjacent or embedded link in order to read about them.
 * 2) For seachability reasons, this page also includes a list of properties.

And always remember that logical conditionals do not allow the converse by default!

How to Read
The theorems are divided into separate tables based on a unifying if statement. Each chart should be used like a map on where you can validly progress in your proof. The tables are divided into three rows: Reference, If, and Then. The first row is devoted to giving you, the reader, some background information for the theorem in question. It will usually be either the name of the theorem, it's immediate use for the theorem, or non-existent. The second row is what is required in order for the translation between one theorem and the next to be valid. The third row is what you can now validly assert as true, without any fear of a contradiction or an invalid statement.

If you come across any numbered lists in this page, that means that all criteria must be satisfied by default. In other words, assume that everything in that list works under AND unless stated otherwise.

Any and all normal naming conventions will be used. For example, ƒ will, by default, refer to a function — unless otherwise specified. This becomes important if certain variable names must be inferred based on context. For example, the "function" L and U for integrals actually represent the lower and upper sum, respectively, and are not necessarily the functions you are used to (so don't apply the function theorems on them!)

List of Axioms
Axioms, logically, are essentially proofs without an if statement. Thus, the following list only contains essentially then statements, which can be used freely.