Recent content by zhang128

Homework Statement If every subsequence of the sequence a_n converges to 0, prove that a_n itself converges to 0. Is the converse true? Prove it or give a counter example. Homework Equations The Attempt at a Solution I can prove a sequence is convergent if it has a convergent...

hmmm, like epsilon/(n^2+n)??

ups, my bad. I was thinking that since rationals are countable, I just need to make the open interval around each rational such that the total sum is less than \epsilon. Then the question turns to prove the sum of a finite series converges to the epsilon??

Homework Statement Prove that the rationals as a subset of the reals can all be contained in open intervals the sum of whose width is less than any \epsilon > 0. Homework Equations The Attempt at a Solution