Sorry it's my first time posting so I am not sure if latex works here... I have been trying to understand the proof of WLLN using Chebychev's inequalities, and here are my problems: I know for the Strong Law of Large Numbers, if n→∞, Ʃ(1 to n) σi ^2 / i^2 < ∞ => SLLN is satisfied, My question is: if n→∞, (n^-2) Ʃ(1 to n) σi ^2 < ∞ (which is a condition of WLLN), does it mean that WLLN is satisfied? It sounds like it is but looking at the proof of WLLN, it doesn't make much sense because the RHS of the inequality is non-zero, thus it does not guarantee that LHS is 0, which is the definition of WLLN. In the Chebychev's inequality (with k=ε/σ), Pr(|sum(Xi)/n - sum(mean_i)/n|≥ε) ≤ σ^2 /ε^2 We know that if the left hand side is 0, then WLLN holds, and if the RHS is 0, then LHS must be 0. Now my problem is if I find that when n→∞ and say σ^2 converges to a non-zero constant (ie finite, so say, RHS becomes 0.5/ε^2), it means that the right hand side is not 0; although this does not imply that the left hand side won't be 0 according to the inequality, I can't think intuitively why the Weak Law will hold. And if finding that RHS is non-zero is not sufficient to show that the Weak Law does not hold, and there is no way of computing LHS since in most cases Xi is random, how would I go about convincing myself that the sequence (does not) satisfies WLLN?