Rudin Principles of Mathematical Analysis Proposition 1.18e Rudin's, Principles of Mathematical Analysis 
Proposition 1.18
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e) If 0 < x < y then 0 < 1/y < 1/x.
Proof:
If y > 0 and v $\leq$ 0, then yv $\leq$ 0. But y $\cdot$ (1/y) = 1 > 0. Hence 1/y > 0.
Likewise, 1/x > 0. If we multiply both sides of the inequality x < y by the positive quantity (1/x)(1/y), we obtain 1/y < 1/x.
Now the part I fail to understand is the placement of the very first sentence. This may seem odd but the introduction of the v seems to be out of place and it's placement in this proof seems arbitrary.
I realize this may seem stupid to ask, but could someone please explain to me why this is in there?

It's being used as a vehicle to prove that $\frac{1}{y} > 0$.
Statement: If $y > 0 $ and $v \leq 0$ then $yv \leq 0$.
Contrapositive: If $yv > 0$ then $v > 0$ or $y \leq 0$.
Using the contrapositive with $v = \frac{1}{y}$ proves that $\frac{1}{y} > 0$.
