The discussion focuses on proving that the set of rational numbers can be contained within open intervals whose total width is less than any given ε > 0. Participants emphasize the countability of rationals, suggesting that by creating open intervals around each rational number, one can ensure the sum of their widths converges to ε. The proposed series for this proof includes ε/(n^2+n) and ε*(1/2)^n, both of which effectively demonstrate convergence to ε.
PREREQUISITESMathematics students, particularly those studying real analysis, and educators looking to deepen their understanding of the properties of rational numbers and convergence in series.
zhang128 said: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??
zhang128 said:hmmm, like epsilon/(n^2+n)??