A standard example of an ind-scheme over a field $\mathrm{k}$ which is not a 
$\mathrm{k}$-scheme is $\mathrm{k}((\varepsilon))$.
My question is how to prove that rigorously? To put it more precisely, 
let $$\mathrm{k}((\varepsilon)) = \{ a \in \prod_{-\infty}^{\infty}\mathrm{k}: 
a_i =0, i \ll 0 \}$$
An ind-scheme is an injective limit of regular schemes. So here, 
$$\mathrm{k}((\varepsilon)) = 
\lim_{i \rightarrow -\infty}\varepsilon^i\mathrm{k}[[\varepsilon]]$$
But why isn't it an algebraic subset of $\prod_{-\infty}^{\infty}\mathrm{k}$?