University of California, San Diego: Edward Bender and S. Williamson's "Arithmetic, Logic and Numbers: Number Theory"
Reread example 2 on page NT-2, with special attention to the proof method, called proof by induction (an exhaustive proof method). This reading discusses a universal statement, which is just a statement that involves universal quantification.
A statement may be provable by evaluating it or by manipulating the symbols using definitions, rules, and theorems to transform the statement to an equivalent statement. Use of truth tables to evaluate a statement is an example of the former. This is also an example of the Method of Exhaustion. If a Universal statement has a finite set for its bound variable (a variable that is quantified is bound and the set of values that it can take is its range or domain), then it can be proven by proving it for each value in the range of the bound variable. If the set is countable, a proof by induction may be applicable. Induction is studied in Unit 4 of this course. Equivalently, the Universal statement can be proved by showing that it is true for an arbitrary value.