I am having a bit of trouble understanding the following statement from SOCR EduMaterials Activities LawOfLargeNumbers - Socr. It suggests that the difference between heads and tails definitely gets bigger (in a coin toss lln experiment)

"...the LLN only guarantees that the sample proportion of heads will converge to the true population proportion (the p parameter that we selected). In fact, the difference |Heads - Tails| diverges! "

I have even seen it written that it increases proportionally witht the square root of n.
I understand that it may increase (and the ratio of success to expected still continues to 0.5) but does it always?

I have run lots of trials using excel (300,000 trials per run) and random walked the difference between heads and expected heads. true, some times the difference tends to increase. sometimes however, it simply fluctuates about zero and there is no discernable trend.

Any thoughts would be appreciated,

Thanks, Dean.