$\lim_{n \to \infty} 2^{n + 2} \times (x^{2^n} - x^{2^{n+1}}) = 0$ For $x \in ]0,1[$, how can one see that:
$$\lim_{n \to \infty} 2^{n + 2} \times (x^{2^n} - x^{2^{n+1}}) = 0$$
Thanks.

Factor $x^{2^n} - x^{2^{n+1}}$ as $x^{2^n}(1-x^{2^n})$, and then the logarithm of the expression is $(n+2)\ln(2) + 2^n\ln(x) + \ln(1-x^{2^n})$.  Clearly this approaches $-\infty$ as $n\to \infty$, (note that  $\ln(x)<0$ and $\ln(1-x^{2^n}) \to 0$) so the original expression tends to $0$.
