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I have to proof that:

$\forall r,c \in \Bbb R$ such that $r\ge 0$ and $c\gt 1$ we have: $$\exists N\in \Bbb N, \forall n\ge N, n^r\lt c^n$$

As I am still struggling with mathematical proofs I wanted to ask if maybe someone could help me with this. I guess I have to do a Mathematical induction here. I also think I have to do a case analysis here but I am not sure about it.

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    "I have to proof" $\to$ "I have to prove" – 2017-01-29

2 Answers 2

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Consider the sequence $f_n=\frac{n^r}{c^n}$. You need to prove that for large enough $n$ this is less than 1. Now consider $f_{n+1}/f_n$ and show that $\exists q<1$ that for large enough $n$ this quotient is less then $q$. That would mean that for large enough $n$ we have $f_{n+k}

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    @user406473 Note: the above is only a sketch of the proof, you should try to fulfill all necessary details - this is a useful exercise. If you fail to do some part, let me know. – 2017-01-29
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Since $$\lim_{n\to\infty}\frac{n} {\log n}=\infty,$$there exists $N$ such that, for all $n\geq N$,$$ \frac{n} {\log n}>\frac r{\log c}.$$ This can be rewritten as $$r\log n