the following is not a true proof, but it shows where the comes from.
By CLT, converges to a (3d) normal r.v.; hence tends to in probability. At this point, it seems that could be turned into . But you need almost-sure convergence, not in probability... And dimension 3 was not yet involved. Note that the result would even hold in dimension 1 and 2 in probability (i.e. with probability tending to 1 for any ), but that it must be false in the almost-sure sense because of recurrence.
Suppose (this is where it gets approximate) that the three components of are independent. If , then each of the three components satisfies , and by the (1-dimensional) CLT, the probability that this bound happens is on the order of where is a standard normal variable. The three components are independent so we get a power 3: .
And at that point you can apply the Borel-Cantelli lemma since the right-hand side gives a convergent series.
(Some of the CLT estimates above only hold in a slightly looser sense, and some small would probably need to be introduced, but I just wanted to give a rough idea of the reason for the . So it comes from which leads to a convergent series)
The real proof should avoid the independence problem... You probably already know some about 3d random walks, so perhaps you know a way? If you want more help, could you please specify if this is a single-question assignment, or if there were questions before? (or what's the context?: what did you already learn about random walks?) And tell us what you tried.