Introduction to Topology and Modern Analysis Problems Thread

I've posted a couple of threads asking for help with the exercises in ITMA in the past month, so rather than continue to clog up the forum with new threads every time I need help I figured I'd just make this thread.

I will not only ask for help in this thread, but will also give assistance to anyone who needs help with a part of the book that I've already worked through. Admittedly, this thread is mostly for my benefit since I'm still on the first chapter, but as I progress I will become more and more able to help others who are trying to work through the book.

So I'm almost finished with the first chapter, but am stuck on the very last problem (Section 8, Problem 10). I feel that I've made it almost to the finish line only to collapse. I've already shown that any uncountable set can be represented as the union of a disjoint class of countably infinite sets, and yet I can't show that U(X_i) has cardinality less than or equal to that of **I**. If someone could give me the nudge that I need to finish this thing off, I'd greatly appreciate it.

Thanks,

Poophead