Section 6.2, due on February 12th

Section 6.2 A More General Principle of Mathematical Induction

What did I find difficult?
The Principle of Mathematical Induction
For a fixed integer m, let S = {i in Z: i > m}  For each integer n in S, let P(n) be a statement.  If

1) P(m) is true and
2) the implication   If P(k) then P(k+1) is true for every integer k in S,

then P(n) is true for every integer n in S.

I'm beginning to understand induction, but I'm still having a tough time with it - mainly with the applications.  It makes sense to me to prove that the smallest one is true, and then to prove the "domino effect" is true.  I understand the concept - if one is true, then the next one is true - but it's hard for me to prove that in different cases.

What did I find interesting?
I really like the thought of induction - I can see how it is very useful.  In looking through the results and their proofs, I can see the genius in applying the induction thought process.  I'm excited to continue learning about it.

I really like the simple proof in result 6.9.

"For every nonnegative integer n, 2^n > n.  We prove that the inequality holds for n=0 since 2^0 > 0.  Assume that 2^k > k, where k is a nonnegative integer.  We show that 2^(k+1) > k+1.  When k=0, we have 2^(k+1) = 2 >1 = k + 1.  We therefore assume that k > k + 1.  

By the Principle of Mathematical Induction, 2^n > n for every nonnegative integer n."

I can see how the simple inductions work.  I am nervous to see how I do with the more difficult inductions.  But I look forward to it.