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?

Its there like a multiplication property of the reals. For if $\frac{1}{y} \leq 0$, then by the property, for $v = \frac{1}{y} \leq 0$, $y\cdot \frac{1}{y} \leq 0$, but $y\cdot \frac{1}{y} = 1> 0$, contradiction.
