OME 1971 Problem 5

Prove that whatever the complex number $z$ is, it is true that $$(1 + z^{2^n})(1-z^{2^n})= 1- z^{2^{n+1}}.$$ Writing the equalities that result from giving $n$ the values $0, 1, 2, . . .$ and multiplying them, show that for $|z| < 1$ holds $$\frac{1}{1-z}= \lim_{k\to \infty}(1 + z)(1 + z^2)(1 + z^{2^2})...(1 + z^{2^k}).$$