Prove that $f(n)\ge 2^{n-1}$

Question

This is from a Brazilian math contest for college students (OBMU):

Let $f: (0,+\infty) \to (0,+\infty)$ be a infinitely differentiable function such that

1. For all positive integer $k$ and positive real $x$, $f^{(k)}(x)> 0$ (where $f^{(k)}$ is the kth derivative).
2. For all positive integer $m$, $f(m)$ is a positive integer.

Prove that $f(n)\ge 2^{n-1}$ for all positive integer $n$.

Attempt

By the mean value theorem, we have

$$f(2)-f(1) = f'(c_1),\space c_1 \in (1,2)$$ $$f(3)-f(2) = f'(c_2),\space c_2 \in (2,3)$$ $$\vdots $$ $$f(n)-f(n-1) = f'(c_{n-1}),\space c_2 \in (n,n-1) $$

Then, $f'(c_k)$ is positive integer for all $k \in \{1,2,\cdots, n-1\}$. Besides, $f'$ is strictly increasing. Thus, $f'(c_k) \ge k$. Adding all the inequalities, we get

$$f(n) \ge \sum_{k=1}^{n-1}k + f(1) = \frac{n(n-1)}{2} + f(1) $$

Answer 1

We have $$f(n+1) = f(n) + \int_n^{n+1} f'(t) \, d t$$ Since $f(n+1) > f(n)$, because $f'(x) >0$, and $f(n+1)$ and $f(n)$ are integers, we see that $\int_n^{n+1} f'(t) \, d t$ is an integer. Define now $$h_1(x) := \int_x^{x+1} f'(t) \, dt.$$ Note that $h'_1(x) = f'(x+1)-f'(x)$. Also note that $h_1(x)$ is integer-valued on $\mathbb{N}$ and has the property that $h^{(i)}_1(x) >0$, i.e. $h_1$ has the same properties as $f$.

We can proceed in this manner to obtain functions $h_n$ with the properties (1) and (2). (For example, apply the same argument on $h_1$ instead of $f$ in order to get $h_2$.) This construction satisfies $$h_i(n+1) = h_i(n) + h_{i+1}(n).$$ We prove by induction that $h_i(j) \ge 2^{j-1}$ for all $i \in \mathbb{N}_0$, whereby we define $h_0 =f$, simultaneously.

1. If $j=1$ then $h_i(1)$ is an integer (for all $i \in \mathbb{N}_0$) and thus the lower bound is satisfied.
2. Assume that the statement was already proven for all $ 1 \le j \le n$ and $i \in \mathbb{N}_0$. Then we have $$h_i(n+1) = h_i(n) + h_{i+1}(n) \ge 2^{n-1} + 2^{n-1} = 2^{n}.$$

As a special case, we get $f(n) = h_0(n) \ge 2^{n-1}$.