Showing $2^{n_2} + 3^{n_3}+\cdots+9^{n_9}$ is dense in $\mathbb{R}^+$ I encountered this problem via a friend. He asked me to prove that 
$$ \left\{u: u= \sum_{k=2}^9 k^{n_k} \quad n_k \in \mathbb{Z}  \right\}$$ 
is dense in $\mathbb{R}^+$.
I was able to show that $0$ is approachable(can get arbitrarily close to) by numbers of this type. However to proceed ahead, I need to know what operations preserve the structure of these numbers. For instance, to show $m+n\sqrt{2}$ is dense for all integers $m,n$, I used the fact that product of any two numbers in this form yields a number in the same form. My problem is I don't know what pattern to exploit in the given problem.
I appreciate any hints / patterns you can provide. One idea I have been working on is as follows:
If we could partition $2^{n_2}, 3^{n_3},\ldots, 9^{n_9}$ into two parts, one that would determine the integer part, the other that would approximate the fractional part, we would be done. 

I don't think this can be done. Think of it like this: Suppose you have $u \in \Bbb{R}_+$ and $u_n \to u$ with
$$ u_n = \sum_{k=2}^9 k^{m_k^n} $$
Now each $(u_n)$ is completely characterized by $(m_2^n,...,m_9^n)$. If $u$ is not zero then not all sequences $(m_k^n)_n$ go to $-\infty$, so some of them are bounded, which means that we will have some values repeated an infinity of times. Then we can extract a diagonal sequence such that every $(m_k^n)_n$ which does not go to $-\infty$ is constant. This means that you approximate $u=2^{a_2}+...+9^{a_9}$ where $a_1,...,a_9$ are integers or $-\infty$ (with convention $a^{-\infty}=0$ for $a>1$.) 
Obviously not every $u$ can be written in this form.
In the same way you can prove that the colsure of the set you mention is precisely
$$ \{ u =2^{a_1}+...+9^{a_9} : a_i \in \Bbb{Z} \cup \{-\infty\}\} $$
