Consider a set of *n* positive real numbers , with for all *i*=1,2,…,*n*. Show then that the inequality

holds.

Our inequality is

.

For *n*=1, both sides of the above reduce to , and thus we have equality.

Now, suppose that the inequality is true for a set of *n* numbers , and let us choose any . Then if we multiply both sides of the inequality by , we get

.

Now, expanding on the right hand side:

since . Thus the equality holds for the set , and so, considering the base case of *n*=1, we have the inequality established by induction.

For a more explicit exemplar of this construction, we apply the above process to the base case:

,

since .

Next,

since .

And so on.

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Tags: Inequality, Math, Monday Math

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