How do I prove all constants to some exponential power belong to little-o of some function I'm trying to prove that c2n = o((loglog n)n) (That's little-o) for any constant c. I understand that we can prove one function grows at a smaller rate than the other by taking the limit as n approaches infinity, and I can very easily pick some arbitrary integer value for c and show that indeed ((loglog n)n) grows at a faster rate. But how do I prove this to be true for any constant c?
Try proving that $5^{2n} = o((\lg \lg n)^n)$.  Write out the proof in detail, and make sure you  understand every step.

Then, try replacing the $5$ in the base with an arbitrary constant $c$.  Does every step still go through?  Hint: you should be able to extend the argument to handle an arbitrary constant $c$, with very little change.

If you're still stuck, edit the question to show us what you've tried, including showing us your proof that $5^{2n} = o((\lg \lg n)^n)$ and your reasoning about which steps are still valid when you replace $5$ with an arbitrary constant $c$.