I'm reading an old book and find the following three question. I'd like to know two things: if my attempts are correct and also it would be great if someone could give suggestions in more detail. Thank in advance
(1) Let $f: \mathbb{N} \backslash\{0\}\rightarrow \mathbb{N} \backslash\{0\}$ be an injective function. Then the series
$$\sum_{n=1}^\infty \frac{1}{(n+1)!} \prod_{k=1}^n f(k)$$
Converges or diverges?
My attempt: (Please someone could help me with this $\prod_{k=1}^n k\ge n!$ I know which is correct but I'm not sure of how prove it.)
$$\frac{1}{n+1}=\frac{n!}{(n+1)!}=\frac{1}{(n+1)!}\prod_{k=1}^n k\le \frac{1}{(n+1)!}\prod_{k=1}^n f(k)$$
Hence by comparison test the series diverges.
(2) Let $m\in \mathbb{N}$ and let $f_m(x)=x^3+3x+m$. Show that $f_m$ cannot have more than two roots if it is restricted to $[0,1]$.
Suppose for sake of contradiction that it has two different roots $a<b \in [0,1]$. Let $c\in (a,b)$. Since in the domain the function is strictly monotonic increasing. Then $f(a)<f(c)$ and $f(c)<f(b)$. But $f(a)=f(b)=0$. Contradiction.
(It can be shown that $f(x)$ is strictly monotonic increasing using the laws of exponentiation. In particular, $a,b\ge 0, r\ge 0$, and $a<b$ then $a^r<b^r$)
(3) Let define the function
$$f(x)=\begin{cases}x^2,& x\in \mathbb{Q}\\ 0,& x\in \mathbb{R} \backslash \mathbb{Q} \end{cases}$$
Let $g(h)=f(h)/h$ when $h\not= 0$. Show that $g(h)\rightarrow 0$ when $h\rightarrow 0$. Let $\varepsilon>0$. We set $\delta=\varepsilon$. So for $0<|x|\le \delta$ we have to show that $|f(x)/x|\le \varepsilon$. If $x$ is irrational the result is trivial. If $x$ is rational (clearly is not zero) then $|f(x)/x|=|x^2/x|=|x|\le \varepsilon$.
Any suggestion it would be great.
In (2) none of the $n$ really affect nothing right?
Question 1: I would choose $f(k) = k+1$, which is clearly injective; for if $f(a) = f(b)$, then $a+1 = b+1$, hence $a = b$. Then $$\prod_{k=1}^n f(k) = 2 \cdot 3 \cdot \ldots \cdot (n+1) = (n+1)!,$$ hence $$\sum_{n=1}^\infty \frac{1}{(n+1)!} \prod_{k=1}^n f(k) = \sum_{n=1}^\infty 1,$$ which is trivially divergent. In fact, the more interesting question is this: for any injective $f$ on the positive integers to the positive integers, is the given sum necessarily divergent?
Question 2: Suppose $0 \le a < b \le 1$ such that $f_m(a) = f_m(b) = 0$. Then $$a^3 + 3a + m = b^3 + 3b + m = 0,$$ or equivalently, $0 = (b-a)(b^2 + ba + a^2 + 3)$. Since $b > a$, it follows that $b^2 + ba + a^2 + 3 = 0$, but this is obviously impossible if $a, b$ are nonnegative. Indeed, the given conditions are much more strict than is necessary: $m$ need not be integer, and the roots need not be bounded above by $1$. In fact, the roots need not be bounded below by $0$, either, but to prove this is a tiny bit more difficult.
Question 3: Your proof is essentially correct as written.