Inequality Problems Using Induction Theorem: Suppose we have a sequence of positive real numbers, a_1, a_2, ..., a_n, such that: a_i > 0 for all i = 1,...
Theorem: Suppose we have a sequence of positive real numbers, a_1, a_2, ..., a_n, such that:
a_i > 0 for all i = 1, 2, ..., n
a_1 + a_2 + ... + a_n = a
Then, a_1 > a.
Proof:
Base Case:
When n = 1, the inequality holds since a_1 is positive.
Therefore, the theorem holds in the base case.
Inductive Hypothesis:
Assume the theorem holds for some n = k. That is, assume a_1 > a_2 > ... > a_k.
Inductive Step:
We need to show that the theorem holds for n = k + 1. That is, we need to show that:
Proof of Inductive Hypothesis:
Using the inductive hypothesis, we have:
Adding a constant a_k to each side, we get:
which simplifies to:
Therefore, the theorem holds for n = k + 1, completing the inductive step.
Conclusion:
By the principle of mathematical induction, the theorem holds for all positive real numbers a_1, a_2, ..., a_n. Therefore, a_1 > a