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? Or maybe the better way to put that would be - why isn't it a $\mathrm{k}$-sub-scheme of a $\mathrm{k}$-scheme $\prod_{-\infty}^{\infty}\mathbb{A}_{\mathrm{k}}^1$?