Not only the theorem you stated does help, but it almost completely proves your assertion
Define You can easily check that the theorem's conditions are fulfilled, hence
Conclude by showing the reversed inequality.
For every family of infinite cardinal numbers on a non-empty index set ,
I do not see how to prove this. We have proved a theorem that says that:
For every indexed family of sets and ever infinite , if and for each , then .
I am not sure if this theorem helps or not though. I need help on this one. Thanks.