ronaldor9 said:
I mean why do you have to start off backwards it seems like in almost all cases where i have used this method (of guessing the inequality and then fixing it to yield a correct statement) all the time the steps are reversible and lead to correct conclusion both ways, i cannot see a time where this would not be correct. (maybe by some division by zero, but i have not encountered such a case)
You have to be careful. I've once made the error myself.
Suppose you have to prove some statement S. Then it is not a valid proof to:
(1) start with S
(2) do some manipulations
(3) and arrive at a true statement.
For example:
(1) Statement S: 5 = 4
(2) Manipulations: 0*5 = 4*0
=> 0 = 0
(3) Since we arrived at a true statement (0 = 0 is true), the statement S is correct.
But the above is of course wrong! This is what HallsOfIvy mentioned.
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The way you start proofs is to write down FIRST what you already know is true, e.g.
we first write down n>=1 (as JG89 mentioned).
It seems a little "magic" and you could ask why I should start with n>=1. Of course, you figured this out by "working backwards" from 1/2 <= n/(n+1). But still, you write down a true statement first and derive something from it.
In mathematics you will often see proofs where they start with some "magical" assumption. For example in delta-epsilon proofs they just start with some delta and you ask yourself how they thought of that delta. Of course, they worked backwards, but you don't include this "working backwards" in your proof.