I've been using mathematical induction to prove propositions like this: $1 + 3 + 5 + \cdots + (2n-1) = n^2$
Which is an equality. I am, however, unable to solve inequalities. For instance, this one:
$ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \leq \frac{n}{2} + 1 $
Every time my books solves one, it seems to use a different approach, making it hard to analyze. I wonder if there is a more standard procedure for working with mathematical induction (inequalities).
There are a lot of questions related to solving this kind of problem. Like these:
- How to prove $a^n < n!$ for all $n$ sufficiently large, and $n! \leq n^n$ for all $n$, by induction? - in this one, the asker was just given hints (it was homework)
- How to prove $n < n!$ if $n > 2$ by induction? Ilya gave an answer, but there was little explanation (and I'd like some more details on the procedure)
- how: mathematical induction prove inequation Also little explanation. Solving it with one line is great, but I'd prefer large blocks of text instead.
Can you give me a more in depth explanation of the whole procedure?