Show that $P>2N \Rightarrow \frac{P}{2}<P-N$.

Question

$N$ and $P$ natural numbers.

If $P > 2N$ then $\frac{P}{2} < P - N$

(sorry for not using the LaTeX)

How do I prove it? i tried but failed. Any help is appreciated. thanks

Answer 1

We have

$$\frac{P}{2} < P - N \iff -\frac{P}{2} < - N \iff -P<-2N \iff P>2N$$

Answer 2

If $P>2N$, then $-P<-2N$ and we have $\frac{P}{2} - (P-N) = -\frac{P}{2}+N < -\frac{2N}{2} + N = -N+N = 0$, from which it follows that $\frac{P}{2} < P-N$.

Answer 3

Divide both sides by $2$:

$$ \frac{P}{2}>N$$

Subtract $P$ from both sides:

$$-\frac{P}{2}>N-P$$

Multiply both sides by $-1$:

$$\frac{P}{2}<P-N$$