Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space. Find a martingale \(\left(u_{n}ight)_{n \in \mathbb{N}}\) for which \(0<\) \(\mathbb{P}\left(u_{n}ight.\) converges \()<1\).

[ take a sequence \(\left(\xi_{n}ight)_{n \in \mathbb{N}_{0}}\) of independent Bernoulli \(\left(\frac{1}{2}, \frac{1}{2}ight)\)-distributed random variables with values \(\pm 1\); try \(u_{n}:=\frac{1}{2}\left(\xi_{0}+1ight)\left(\xi_{1}+\xi_{2}+\cdots+\xi_{n}ight)\).]