I want to prove the following; $\forall t>0\ \ \forall m\in \mathbb{N} \ \ \exists N \in \mathbb{N} \ \ \forall n\geq N: \ (1+t)^n > n^m.$
For readers who hate quantifiers, here's the version in words: "$(1+t)^n$ eventually surpasses $n^m$ for some $t>0$ and $m\in \mathbb{N}$".
Though this sounds simple enough, I couldn't manage to prove it using only very elementary statements about the real numbers, like the archimedian property etc. (so no derivates and so on involved).
My questions are:
1) Is there a simple (as described above) proof for this ? (If there isn't, I would also be happy with a proof using more of the analysis machinery.)
2) Is there a way to express the least $N$, which satisfies the above, in a closed form ?
3) I have somewhere heard of a theorem, that two real convex functions can have at most two intersection points (and the above statement seems closely related to this theorem), so I would be very happy, if someone could also give me a reference for this theorem.