##
Proof by cases

The idea in proof by cases is to break a proof down into two or more
cases and to prove that the claim holds in every case. In each case, you
add the condition associated with that case to the fact bank for that case only.
As long as the cases cover every possibility, you have proved the claim
regardless of what the actual case is.
Proof by cases is closely
related to the idea of using If-statements in a computer program.

### Example

**Theorem.** For every integer *n*, *n*^{2} ≥ *n*.

**Proof.**. By cases. There are three cases:
*n* ≤ −1, *n* = 0 and *n* ≥ 1.

**Case 1. (***n* == 0). Notice that 0^{2} ≥ 0, so the theorem
holds when *n* = 0.

**Case 2. (***n* ≥ 1).

(1) |
*n* ≥ 1 |
(from the condition for this case) |

(2) |
*n*^{2} ≥ *n* |
(multiply both sides of (1) by the positive value *n*) |

So the theorem holds in this case.
**Case 3. (***n* ≤ −1).
Since *n* ≤ −1 and *n*^{2} ≥ 0, the
theorem clearly holds in this case.