Show that as positive real numbers get smaller their inverse gets larger.

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Problem: 

Show that as positive real numbers get smaller their inverse gets larger.

Answer: 

SAMPLE SOLUTION

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Solution: 

Let's say we have <Sign in to see all the formulas>. Since <Sign in to see all the formulas> is never zero and is always positive we can divide to find that

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Now what if we have <Sign in to see all the formulas> such that <Sign in to see all the formulas>? Again, because <Sign in to see all the formulas> and <Sign in to see all the formulas> are never zero and are always positive we can divide to find that

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Now putting the inequalities together we see that

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We can do this for more values of <Sign in to see all the formulas> to find that

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So as <Sign in to see all the formulas> gets smaller and smaller (i.e. closer to zero), <Sign in to see all the formulas> gets bigger and bigger.