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I have taken the proof approach from some previous problems in Spivak's book on Calculus (3rd edition).

This is problem 5.(viii) in chapter 1: Basic Properties of Numbers.

I did as follows:

If [itex] a = 0 [/itex] or [itex]c = 0[/itex], then [itex]ac = 0[/itex], but since [itex]0 < bd [/itex], so [itex]ac < bd [/itex]

Otherwise,

[itex]0 < a [/itex] & [itex]b > a[/itex] & [itex]0 < c[/itex] & [itex]d > c [/itex]

Therefore,

[itex]a \in \mathbb{R}[/itex] and [itex] (b-a) \in \mathbb{R}[/itex] and [itex]c\in \mathbb{R}[/itex] and [itex] (d-c) \in \mathbb{R}[/itex]

[itex]a(b-a)\in \mathbb{R}[/itex] and [itex]c(d-c)\in \mathbb{R}[/itex]

[itex]a(b-a)[/itex] and [itex]c(d-c)\in \mathbb{R}[/itex]

This is where I am stuck.

From here, How can I algebraically deduce that [itex]ac < bd [/itex]?

Any hint, would be greatly appreciated,

Thank you very much in advance.