I understand the differences between the two concepts, but they look similar so I was searching for some theorems which tie them. I found that a sufficient condition for an estimator $T_n$ to be consistent is to be asymptotically unbiased and such that $\lim_{n} \operatorname{Var}(T_n(X))$ (the source is: https://it.wikipedia.org/wiki/Consistenza_(statistica), it's the italian version of the Wikipedia page about consistency, I didn't link the English version since that statement is not present there).
No proof is given there and I'd like to know if this theorem is actually correct and if anyone could help me proving that.
Also, I know that consistency doesn't imply asymptotic unbiasedness, but I haven't found a good example for understanding it more clearly.
