Prove that is the empty set.

By definition, is the set of functions such that for any positive constant and all . is the set of functions such that for any positive constant and all .

So, is the set of functions such that . Now, this inequality cannot be true asymptotically as becomes very large, cannot be simultaneously greater than and less than for any constants . Hence, no such exists.

Another way to look at this is By definition,

and

Both of this cannot be simultaneously true. Hence, no such exists.

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