Are there particular cases of Muntz theorem which can be proved with an elementary way? The Muntz theorem states that the space $S := span\{x^{\lambda_i}\}_{i \geq 0}$ is dense in $(C[0,1], \| .\|_{\infty})$ if and only if $\sum_{i = 0}^{\infty} \frac1{\lambda_i} = \infty$ (where $\lambda_0 = 1$ and {$\lambda_i\}_{i \geq 1}$ are stricly positive real numbers).
I'm interested in the implication : $\sum_{i = 0}^{\infty} \frac1{\lambda_i} < \infty$ implies $S$ is not dense. The proof I know uses complex analysis (with infinite product) and Hahn-Banach theorem.
But, are there particular cases (for example : $\lambda_n = n^2$ or $\lambda_n = n!$) where this implication can be proved in a more elementary way ?

We can prove it simply by :
1- first considering the space $(C[0,1], \|. \|_2)$ where we can compute explicitly the $L^2$ distance between $x^m$ and $span(x^{\lambda_1}, \dots, x^{\lambda_n})$. It is given by determinant of a gram matrix. Thus we prove the result in $(C[0,1], \|. \|_2)$.
2- then for the case $\|. \|_{\infty}$ to approximate a continuous function $f$, we approximate $f$ by a $C^1$ function $g$ (a polynomial for example) and then approximate $g'$ in $L^2$ norm.
