<s>[INST] Prove that if $a$ and $b$ are real numbers with $0 < a < b$ then $\frac{1}{b} < \frac{1}{a}$ 
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? 
 [/INST] 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.  
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