Re: Mark Bridger's request for better induction\well-ordering examples.
The three "archetypes" found in most intro-to-proof texts are 1. Summation\product formulas. 2. Inequalities. 3. Divisibility propositions. Some are useful, most are at least historically interesting. As already pointed out, lack of discovery is common to these examples.
Good sources: _A Transition to Advanced Mathematics_, Smith-Eggen-St. Andre_ _Chapter Zero_, Carol Schumacher _Introduction to Mathematical Structures_, Steve Galovich
Another collection of very interesting examples have origin in general position arguments and discrete mathematics. Here are a couple of examples:
1. Every partition of the plane by a finite number of straight lines can be two-colored. 2. In a round-robin tournament (no ties) a _top player_ is a player#1 who, for every other player#2, either beats that player#2 or beats some other player#3 who beats that player#2. Every such tournament with a finite number of players has at a top player. 3. Any 2^n-by-2^n chessboard with one deleted square can be tiled using 2-by-2 with one square deleted tiles. 4. A jig-saw puzzle is assembled by successively joining pieces together into blocks. A move is the joining of two pieces to form a block, the joining of a piece to an existing block, and the joining of two exist- ing blocks. Every n-piece puzzle requires exactly n-1 moves.
Although not calculus. I will post this to the group on the odd chance that this of interest to others than just Mark.