Proofs: Problem 4

Before we reveal the correct solution, compare your answer to the wrong solution below. Did you put your statements into logical order? If not, fix this before proceeding.


When we think about this, it is natural to start with what is to be proved: \(\frac{2pt}{p+t} \ge \frac{2pr}{p+r}\), and to algebraically manipulate it until we get to the known \(t \geq r\). This is backwards, though instructive, so let's see how it goes:

\(\frac{2pt}{p+t} \ge \frac{2pr}{p+r}\)

Multiplying each side by (p+t)(p+r) gives us \((2pt)(p+r) \geq (2pr)(p+t)\)

Multiplying out both sides gives us \(2p^2t + 2ptr \geq 2p^2r + 2ptr\). Subtracting 2ptr gives \(2p^2t \geq 2p^2r\). Dividing by \(2p^2\) (which is positive) gives us \(t \geq r\). We know this is true, so we are done.

This is NOT a proof!!

A direct proof for an implication \(p \rightarrow q\) starts by assuming p, and then proceeds step-by-step to conclude q. Here, we've worked backwords. While this is a good way to think about why the statement is true, it is not a valid to prove it. We need to flip the proof around, and start with the assumption \(t\geq r\), and then conclude that \(\frac{2pt}{p+t} \geq \frac{2pr}{p+r}\).

Often you can fix proofs like this by writing a new draft with your work reversed.

Checking a bit more

Now that your work is in forwards order, make sure you written it up properly. Did you declare your variables at the start and then assume the hypothesis of the claim? Does your proof end at the conclusion of the theorem (not one step before it)?

Is it written up nicely, with connector words and sentences? You'll have to do this on an exam, so practice doing it now.

Then look at the model solution.