Null Sequences (II) I came across the following problems on null sequences during the course of my self-study of real analysis.

Let $x_n = \sqrt{n+1}- \sqrt{n}$. Is $(x_n)$ a null sequence?

Consider $y_n = \sqrt{n+1}+ \sqrt{n}$. Then $x_{n}y_{n} = 1$ for all $n$. So either $(x_n)$ or $(y_n)$ is not a null sequence. It seems $(y_n)$ is not a null sequence. I think $(x_n)$ is a null sequence because $\sqrt{n+1} \approx \sqrt{n}$ for large $n$ which implies that $x_n \approx 0$ for large $n$. 

If $(x_n)$ is a null sequence and $y_n = (x_1+ x_2+ \dots + x_n)/n$ then $(y_n)$ is a null sequence.

Suppose $|x_n| \leq \epsilon$ for all $n >N$. If $n>N$, then $y_n = y_{N}(N/n)+ (x_{N+1}+ \dots+ x_n)/n$. From here what should I do?

If $p: \mathbb{R} \to \mathbb{R}$ is a polynomial function without constant term and $(x_n)$ is a null sequence, then $p((x_n))$ is null.

We know that $|x_n| \leq \epsilon$ for all $n>N$. We want to show that $|p(x_n)| \leq \epsilon$ for all $n>N_1$. We know that $p(x) = a_{d}x^{d} + \cdots+ a_{1}x$. So $$|p(x_n)| \leq a_{d} \epsilon^{d} + \cdots+ a_{1} \epsilon$$
for all $n>N_1$. 
 A: You have made pretty good progress on all three problems.
For problem 1: note that in fact 
$\lim_{n \rightarrow \infty} y_n = \lim_{n \rightarrow \infty} \sqrt{n+1} + \sqrt{n} = \infty + \infty = \infty$, 
so 
$\lim_{n \rightarrow \infty} x_n = \lim_{n \rightarrow \infty} \frac{1}{y_n} = \frac{1}{\infty} = 0$.
For problem 2: you have
$y_n = y_N (N/n) + (x_{N+1} + \ldots + x_n)/n$.
It's enough to show that both terms of the right hand side can be made arbitrarily small as $n$ gets arbitrarily large.  The first term is a constant divided by $n$: this goes to zero with $n$.  The second term is a sum of at most $n$ things each one of which is in absolute value at most $\frac{\epsilon}{n}$, so the sum is in absolute value at most $\epsilon$.  So you're basically done.
For problem 3: If you choose $\epsilon \leq 1$ then $\epsilon^n \leq \epsilon$ for all $n \geq 1$, so 
$|a_d \epsilon^d + \ldots + a_1 \epsilon| \leq |a_d + \ldots + a_1| \epsilon$, a quantity which goes to zero with $\epsilon$.
A: Try this.
$$x_n = {(\sqrt{n+1}-\sqrt{n})(\sqrt{n+1} + \sqrt{n})\over \sqrt{n+1} + \sqrt{n}}
= {1\over \sqrt{n+1} + \sqrt{n}}.$$
This shows $x_n\to 0$ as $n\to\infty$.
