Monday Math 69

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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