I am currently reading the proof of the Müntz-Szász theorem for $L^2[0, 1]$ on page 6 of this article and I'm stuck at this point:
We have a sequence $(\lambda_i)_{i=1}^\infty$ of real numbers with $\lambda_i > -1/2$ for every $i$ and a fixed natural number $m$ different from every $\lambda_i,$ and the authors claim that $$ \sum_{\substack{i= 1 \\ \lambda_i > m}}^\infty \frac{2m+1}{m+\lambda_i+1}=+\infty \Longleftrightarrow \sum_{\substack{i= 1 \\ \lambda_i > m}}^\infty \frac{1}{2\lambda_i + 1} = +\infty.$$
Now, I see that $\Longleftarrow$ is true, because $$\frac{2m+1}{m+\lambda_i+1} \geq \frac{2m +1}{2\lambda_i +1}$$ for all $i$ with $\lambda_i > m,$ but I haven't been able to prove the other implication. I'd appreciate any help on this.