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 dont 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