Prove that if $a$ and $b$ are real numbers with $0 < a < b$ then $\frac{1}{b} < \frac{1}{a}$

Question

Suppose a and b are real numbers. Prove that if $0 < a < b$ then $\frac{1}{b} < \frac{1}{a}$.

My attempt:

Given that $0<<$

We can write $b$ as $b = an$, where $n>1$ $$\tag1\frac{1}{a} = \frac{n}{an} = \frac{n}{b} $$ $$\tag2\frac{n}{b} > \frac{1}{b} \implies \frac{1}{a} > \frac{1}{b}$$

Is it correct?

Answer 1

Yes, your solution is correct.

Of course you could have multiplied both sides of your inequality by $$\frac {1}{ab}$$ to get the result in one shot.