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I was just introduced into elementary proofs of inequalities, my text's explanation however feels incomplete. I did further research on the subject, my question is thus:

Prove: If $0 < a < b$, if $c < d$, and $c > 0$, then:

$$ac < bd$$

I understand that you may add or multiply an inequality by a number; however i cannot seem to determine what to use to show what is desired. Another approach i attempted is the fact that if $a < b$ then there is some k s.t. $a + k = b$. And similar $m$ for $c < d$. However plugging this in for $ac < bd$ proved no avail.

What are some ways i could solve this? I am sure there is more than one way.

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    $\begingroup$ If $0 < a < b$, if $c < d$ and $c>0$ Let $\mathcal{P}^{+}$ be the set of positive numbers. Since $(b-a), c \in \mathcal{P}^{+}$, then $c(b-a)\in P^{+}$, i.e., $ac<bc$. Also $(d-c),b\in \mathcal{P}^{+}$ so $b(d-c)\in \mathcal{P}^{+}$,i.e., $bc<bd$. Then we have $ac<bc$ and $bc<bd$ and hence (by transitivity) $ac<bd$ as desired. $\endgroup$ Mar 28, 2014 at 2:17

4 Answers 4

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Given: $$\begin{equation}\tag{P1}0 < a < b\end{equation}$$$$\begin{equation} \tag{P2} 0 < c < d \end{equation}$$$$c\times (P1) \implies \begin{equation}\tag{3} ac < bc\end{equation}$$$$b\times (P2)\implies \begin{equation}\tag{4}bc < bd\end{equation}$$$$(3)\land(4)\implies\begin{equation}\tag{C} ab < bc < bd\end{equation}$$$$\therefore ac<bd$$

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Using

  • if $0 \lt y $ then $x \lt x+y$
  • if $x \lt y $ then $0 \lt y-x$
  • if $0 \lt x$ and $0 \lt y $ then $0 \lt xy$

you have

$$ac \lt ac+ (b-a)c = bc \lt bc+b(d-c)=bd$$

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  • $\begingroup$ You can multiple/add sides of an inequality with a different number? I got the impression such an operation must be the same across the entire inequality $\endgroup$
    – user121947
    Mar 28, 2014 at 1:52
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    $\begingroup$ This is four inequalities and equalities on one line. If you prefer: $$ac \lt ac+ (b-a)c$$ $$ac+ (b-a)c = bc$$ $$bc \lt bc+b(d-c)$$ $$ bc+b(d-c)=bd$$ so $ac \lt bd$ $\endgroup$
    – Henry
    Mar 28, 2014 at 7:13
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Start with $a<b$. Multiply both sides by the positive $c$ and we get $ac<bc$, and since $c<d$ and $c$ and $d$ are both positive ($c$ is positive, and $d$ is larger than $c$, so we can tell that $d$ is positive),we can tell that $ac<bc<bd$, an we can simplify that as $ac<bd$.

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There was nothing wrong with your approach as well.

$\begin{aligned}a < b & \implies a + k = b &\text{ for some } k > 0 \\ c < d & \implies c + m = d & \text{ for some } m > 0\end{aligned}$

The desired inequality then becomes prove that:

$\begin{aligned}& ac & < & bd \\ \text{i.e } & ac & < & (a+k)(c+m) \\ \text{i.e } & ac & < & ac + am + kc + km \\ \text{i.e } & 0 & < & am + kc + km\end{aligned}$

which is true, since each term on the RHS is positive ($a, c, k, m > 0$).

Replacing $x, y$ with $x, x + \alpha$ where $\alpha = y - x \ge 0$ is often a useful strategy in proving inequalities. See this for example.

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