1. The problem statement, all variables and given/known data Construct a sequence that has all rational numbers in it 2. Relevant equations None. 3. The attempt at a solution Here are my thoughts, though I have no solutions yet. If I construct a sequence Sn= n*sin(n)-1/n, will it work? Thanks guys!
The sine term will often give irrational numbers, so that won't work. Try putting the rationals in an array and finding a path that goes through all of them.
thanks! can you elaborate a little bit? I'm trying to self-study real analysis, and I'm not really familiar with what you just mentioned... how would i put them in an array and find a "path"? thanks!
Just drop the sine term. OR Do you really want a sequence with all of the rational numbers in it. -- maybe just all of the positive rationals? Better yet: Please type the problem word fro word as it was presented to you.
Hi guys: Thank you so much! Here's the problem as it was typed on the book: Construct a sequence such that every real number is its limit point. I know this is different from the question I typed above, but my reasoning is that if i can have a sequence that contains all rational numbers, then I can prove that every real number is its limit point. Does that make sense? How should I solve the original question if this does not? Thank you!
You might look here for an idea. http://www.homeschoolmath.net/teaching/rational-numbers-countable.php