In most students' introduction to rigorous proof-based mathematics, many of the initial exercises and theorems are just a test of a student's understanding of how to work with the axioms and unpack various definitions, i.e. they don't really say anything interesting about mathematics "in the wild." In various subjects, what would you consider to be the first theorem (say, in the usual presentation in a standard **undergraduate** textbook) with actual content?

Some possible examples are below. Feel free to either add them or disagree, but as usual, keep your answers to **one suggestion per post.**

- Number theory: the existence of primitive roots.
- Set theory: the Cantor-Bernstein-Schroeder theorem.
- Group theory: the Sylow theorems.
- Real analysis: the Heine-Borel theorem.
- Topology: Urysohn's lemma.

Edit: I seem to have accidentally created the tag "soft-questions." Can we delete tags?

Edit #2: In a comment, ilya asked "You want the first result *after all the basic tools have been introduced?*" That's more or less my question. I guess part of what I'm looking for is the first result that justifies the introduction of all the basic tools in the first place.

`example = examples`

and`soft-question = soft-questions`

(and`topos = topoi`

:) ). $\endgroup$ – Ilya Nikokoshev Oct 24 '09 at 20:34`object = objects`

idea seem to be easy to implement to me -- is there an example where this merging makes no sense? $\endgroup$ – Ilya Nikokoshev Oct 24 '09 at 23:20