Several Questions related to Convergence and Cauchy Sequences 1) a sequence ($x_n$) of irrational numbers that converges to a rational limit
I pick $\pi$/n....I am wondering is $\pi$/n an irrational number since rational number is any number can be written in fraction form.
2) a sequence ($x_n$) of rational numbers that converges to an irrational limit
My guy feeling tells me this is possible...however I am not able to find an example...does anyone think of one?
3) a sequence ($x_n$) containing subsequences converging to every number in N
I know 1,1,1,2,1,2,3,1,2,3,4,1,2,3,4,5 works....I am wondering does N work? If not, why not?
4) a sequence $(x_n)$ that is not Cauchy, but for which $|x_{n+1} - x_n|$ converges to $0$
I am told a sequence of partial sums $x_{n}\sum_{i=1}^{n}\frac{1}{i}$ works. However, doesn't this partial sum converge to 1 since  $x_{n+1} = 1/2+1/3+1/4+1/5+...+1/(n+1)$ and $x_n = 1+1/2+1/3+1/4+1/5+...+1/n$? If we subtract $x_n$ from $x_{n+1}$, we get $1/(n+1) - 1$ which is $-n/(n+1)$. Then doesn't $|x_{n+1} - x_n|$ converge to 1?
 A: Regarding questions $1$ and $2$:
$\{s_n\}=\{2,2.7,2.71,2.718,\ldots\}$
The sequence is made up of rational numbers, but tends to $e\in\mathbb{R}\setminus\mathbb{Q}$, as $n\to\infty$.
$\{s_n\}=\{\dfrac{\pi}{n^3}\}$
The sequence is made up of irrational numbers, but tends to $0\in\mathbb{Q}$, as $n\to\infty$.
Of course $\{\dfrac{\pi}{n}\}$ would be valid as well.
A: For (1), you're wrong at "rational number is any number can be written in fraction form". That's not quite true; a rational number is one that can be written as a fraction of integers. If simply being the result of a division were enough, then every number would be rational, because any $x$ can be written as $\frac x1$, and then the concept of "rational number" wouldn't be worth much.
For (3), if by "does N work" you mean you're suggesting the sequence
$$1,2,3,4,5,6,\ldots $$
then that doesn't work. For example you can't find any subsequence that converges to $1$. Since the indices that make up a subsequence must, by definition, be strictly increasing, the $k$th term of any subsequence cannot come from an earlier term than the $k$th term of the original sequence. And that means that every term except for the first of any subsequence of the above sequence must be $\ge 2$, so there's no way the terms of the subsequence can be arbirtarily close to $1$ late in the subsequence.
