The problem is to show that $P(|X_n| \ge c \sqrt{\ln n}\text{ i.o.}) = 0$ for standard normal $X_n$ that are not necessarily independent. Also, identify the smallest such $c$.
I am thinking that the Borel-Cantelli lemmas are the way to go here but I can't figure out how to get a bound on this probability. Chebychev's inequality does not work. I suppose I will need to use Markov's inequality and the 4th moment. But even in that case how would I find the smallest $c$? Any suggestions?