Quiz #6 2008/04/15 18:12
Q. Prove that if
\{x_n\} is a sequence which satisfies|x_n|\le\frac{1+n}{1+n+2n^2}for all n\in\mathbb N, then x_n is Cauchy.A.
뭥미? 하아...
|x_n|\le\frac{1+n}{1+n+n^2+n^2}\le\frac{1+n}{1+2n+n^2}=\frac{1}{1+n}So, for every \varepsilon > 0, there exists N\in\mathbb N such that m,n\ge N implies |x_m|<\frac{\varepsilon}{2} and |x_n|<\frac{\varepsilon}{2}. then|x_m-x_n|\le|x_m|+|x_n|<\varepsilon.So \{x_n\} is a Cauchy sequence.
