Set up the proof by finding a "base case" and an "inductive step." The basic idea is to find a small number for which the proposition is true and then find a statement that ensures that if the proposition is true for one number it is true for the next higher number. For example, suppose you are trying to prove that the sum of the numbers from 1 to n is equal to n(n+1)/2. It is certainly a statement about an infinite set of things --- all the numbers. The base case is for n equals a small number. For example n = 1. The inductive step is "if this statement is true for n it is also true for n+ 1."
Find the base case, and prove that it is true. For example, if you are trying to prove that the sum of the numbers from 1 to n is n(n + 1/2, you first prove it for a small number like n = 2. If N = 2 the statement becomes "the sum of all the numbers from 1 to 2 is (2)((2) + 1)/2." This statement is " 1 + 2 = (2 X 3)/2" which is true.
State and prove the inductive step. If you are trying to prove that the sum of the numbers from 1 to n is n(n + 1)/2, the inductive step is "if the sum of the numbers from 1 to n is n(n + 1)/2 then the sum of the numbers from 1 to (n + 1) is (n + 1)((n + 1) + 1)/2." Notice that the sum of the numbers from 1 to (n + 1) is n(n + 1)/2 + (n + 1) and (n + 1)((n + 1) + 1)/2 = (n + 1)(n + 2)/2 = ((n + 1)n + (n + 1)2)/2 = n(n + 1)/2 + (n + 1), so the proposition is proven.