Let Xnn be i.i.d. simple random variables with mean 0 and variance 1. Then law of iterated logarithms holds. Set Sn=X1+⋯+Xn. From P(lim sup and P bigg( liminf_n fracS_n sqrt2n log log(n) = -1 bigg) = 1,together with the uniform bounded-ness of the X_n, deduce that with probability 1 the set of limit points of the sequence bigg fracS_n sqrt2n log log(n) bigg is the closed interval from –1 to + 1. I am not sure where to start. Assumptions tell you where you the smallest and largest the limit can be, but takeng x in [-1,1], how do I build a sequence of the form x_k _k = bigg fracS_n_k sqrt2n_k log log(n_k) bigg _k such that x_k to x? Maybe this is not a good way to approach the problem.