Let $(u_n)$ be a sequence such that $u_n^2 = 1$ for all naturals $n$.
Define a sequence $(v_n)$ by $ v_n := \displaystyle\sum_{k=0}^n \dfrac{u_0\times u_1\times \cdots \times u_k}{2^k} $.
It is asked to prove that $(v_n)$ converges and that the square of its limit equals $4$.
For the first part, It's not hard to prove that $(v_n)$ is Cauchy thus it converges.
Any ideas, hints for the second part ?
thanks.