Suppose the joint limit of {xn,m}n,m=1∞ is L, and suppose that for all n, limm→∞xn,m exists, and for all m, limn→∞xn,m exists. Then show limn→∞limm→∞xn,m=limm→∞limn→∞xn,m=L.
(Recall: L is the joint limit of {xn,m}n,m=1∞ if for every ϵ>0, there exists an M such that if n≥M and m≥M, then ∣xn,m−L∣<ϵ.)