no the lln is a much stronger result than this it says that you are likely to get close to ones in rolls of a die this is the fundamental thing that makes statistics worthwhile if you have a lot of observations then the average of the sample is close to the average of the population this is why you can eg do clinical trials in only a few thousand people and extrapolate to approving drugs for a population of hundreds of millions i agree that the article could use revision in the introduction i don't really like the proof either only a very small number of people will understand it and for them there is a simpler shorter proof under similar assumptions by algebra the variance of a sum is proportional to the number of summands and so the variance of an average decreases as the number of summands increases tsl