For ten points, identifythis doubly-eponymous theorem which states that every bounded sequence has a convergent subsequence.
This statement was  rst proved as a lemma on the way to proving the intermediate value theorem. This theorem can be easily derived from Cantor's intersection theorem, which deals with nested intervals. One way of stating this theorem is that every in nite bounded set in Rn has an   accumulation point. A corollary of this theorem is the Heine-Borel theorem, that is, that any subset of Rn is compact if and only if it is closed and bounded. For ten points, identifythis doubly-eponymous theorem which states that every bounded sequence has a convergent subsequence.
This statement was  rst proved as a lemma on the way to proving the intermediate value theorem.
Bolzano-Weierstrass theorem [accept sequential compactness theorem as well as descriptions of its statement until  doubly-eponymous  and prompt afterward]