For some positive integer $n$, let $x_1, \dots, x_n$ be positive real numbers. Consider the set of all linear combinations of $x_1, \dots, x_n$ with nonnegative, rational coefficients:
$$ LC= \left\{ \sum_{i=1}^n q_i x_i \; : \; q_i \in \mathbb{Q} \cap[0, \infty) \right\}$$
For any finite sequence of positive real numbers $(x_1, \dots, x_n$), I think there is always an irrational number $r$ such that $r \notin LC$. Moreover, this unattainable irrational number can be made arbitrarily large. How does one prove this?
Do we ever need irrational coefficients to generate a rational? <-- this might answer my question, but I do not understand it, and I wonder if there is a simpler solution.
The question can be reformulated: let $f : \mathbb{Q^n} \to \mathbb{R}$ be a function of the form $f = \sum_{i=1}^n \lambda_i q_i$ for $\lambda_i \in [0, \infty)$ and variables $q_i \in \mathbb{Q}$ (this is slightly more general), $i = 1, \dots, n$. Then there exist arbitrarily large values $r \in \mathbb{R} \setminus \mathbb{Q}$ such that $r \notin f(\mathbb{Q}^n)$.