Proof by ??? Question

Question

I have been given a question to do as part of a task set by our lecturer, however he has failed to provide any examples of similar questions with all the corona stuff going on. Here is the question

Suppose that v ∈ N

Prove that: If v ≥ 2 then v^2+ 1 > 2v − 1

I am really struggling to figure it out and I would be really grateful for some help. I have a lot of similar questions and if I could get an answer for this one then I could do the rest pretty easily.

Thanks

Answer 1

Since this involves an inequality, we're going to have to prove this for 2 cases: case #1 for v = 2 and case #2 for v › 2. For the first case, this is pretty easy since we simply insert 2 for v and do the arithmetic: 2² + 1 › 2·2 1, so we have 5 › 3 and this proves case #1.

For case #2, we need to use an expression that will cover ALL values that would be greater than 2, so let's use an expression like v + h, where h > 0, v = 2, and so v + h is guaranteed to be bigger than 2. Now if we insert v + h into the original expression, we get ( v + h )² + 1 > 2(v+h) - 1. If we move all the "v+h" terms to the left side and move the +1 to the right side, we get ( v + h )² - 2(v+h) > -2. If we complete the square on the left side, treating the (v+h) the same way we would if it was "x", we would have ( v+h)² -2(v+h) + 1 > -2 + 1. If we now factor the left side as a Perfect Square Trinomial, it becomes [( v+h ) - 1)]² > -1. Since any squared real number has to be a positive number, the left side is definitely greater than -1 and this proves case #2.

Answer 2

There's more than one way to do this. Are you doing induction proofs in your class? In case the answer is yes:

Basis step. For v = 2 we have 2² + 1 ?>? 2(2) - 1, which is true since 5 > 3

Induction step. Assume true for some value v ≥ 2: (*) v² + 1 > 2v - 1

For v + 1: (v + 1)² + 1 ?>? 2(v + 1) - 1

(**) v² + 2v + 1 ?>? 2v + 2 - 1

Compare the two inequalities: To get from (*) to (**), we

1. Add 2v to the left hand side
2. Add 2 to the right hand side

The amount we added on the left hand side is larger than the amount we added on the right hand side.

The left hand side of (*) was already larger than the right hand side of (*)

Therefore the left hand side of (**) is larger than the right hand side of (**)