## 2009年12月11日 星期五

### Two Analysis Problems

Recently I heard two problems in analysis, both I think are interesting, and they do not require too deep knowledge in analysis, which is the kind of questions I like most. Share here.

1) Suppose $\sum_{j=1}^\infty |a_j|$ converges. Also, for each positive integer k, it is known that $\sum_{j=1}^\infty a_{jk} = 0$ (just to avoid confusion, allow me clarify here that jk means "j times k"). Prove that $a_i=0$ for all positive integers i.

2) S contains all elements $x\in[0,1]$ such that for any $\epsilon>0$, there exists a rational number $\frac{p}{q}$ (where p,q are positive integers) satisfying $\left| x - \frac{p}{q}\right| < \frac{\epsilon}{q^3}$. Prove that S is uncountable.

#### 3 則留言:

Singmay 提到...

For the first problem, it is even more interesting to ask whether there are non-zero examples when the assumption on absolute convergence is removed. Can someone give such an example?

Polam 提到...

For the second one, it follows from Baire Category Theorem, right?

Marco_Dick 提到...

To Singmay: just seen your comment, thinking...

To Polam: I did not realize that Baire Category Theorem is useful here. I just use elementary approach to show that S contains a subset which is equivalent to 2^N, which is uncountable.