I have switched for a moment from calculating integrals and solving differential equations to learning how to prove statements about infinite sets.
I have made a list of exercises for training and found myself absolutely helpless even in the face of the simplest problems in that list. The thought process that is required, currently, is alien to my brain, so I want to start with a little help from the community.
The problem is:
Given an uncountable set A of positive numbers, prove that we can choose a countable subset B from set A, so that the sum of the elements of B is infinite.
I guess, we should somehow provide the construction of the set $B \subset A$ with the required property, but I don't event know how to start.
I'd like to see the thinking process with as much detail as possible, so that I could internalize this type of thinking.
No, my question is not duplicate. Why do you mark it a duplicate so fast?
1. I need to extract countable set B, not any set.
2. I am more concerned with the specific type of thinking that is needed to solve these kinds of problems, than particular one-line formula that does it. In order to make my question concrete, I have provided a specific problem. By explaning how we are coming up with the solution to that problem it is easy to show how one thinks in the process.
A lot of interesting and useful information has been generated in the comments :) Now I am going into analysing mode and start putting this information together inside of my brain (and mold it with additional thinking). It is interesting, what time will it take before I get a general feeling for what is happening :)