Cardinality of finite , infinite sequences and bounded function

Question

What is the cardinal number of these sets :

$1.$ The set of finite sequences that its items are taken from inifinite sequences of $\mathbb Z.$

$2.$ The set of infinite sequences that its items are taken from finite sequences of $\mathbb R.$

$3.$ $A=\left\{f: \mathbb R_+ \to \mathbb R_+ | f(x)\leq e^{-x} \right\} $

$4.$ $C:=\left\{h \in \left\{ 0,1 \right\}^\mathbb N| \frac{1}{n+1} \sum_{k=0}^{n} h(k) \leq \frac{1}{2} \right\} $

My solution:

$1.$ I know that |Infinite sequence($\mathbb Z$)|=$|\mathbb Z^\mathbb N| =
\mathfrak{c}$
but i dont know how to deal with finite sequence of infinite sequence($\mathbb Z$).

$2.$ I know |inifnite sequence of finite sequence($(\mathbb R ^n)^ \mathbb N$)|=|$\mathbb R^n|^{|(\mathbb N)|}=|\mathbb R^{|(\mathbb N)|}|=\mathfrak{c}$.

$3.$ I have no idea how to deal with this except High limit is $|\mathbb R|^{|\mathbb R|} = 2^{\mathfrak{c}} $

$4.$ High limit is $\mathfrak{c}$ but i cant find low limit.

Is my proof correct for 2 correct ? Please help in Exercises 2,3,4.

Answer 1

I don't know your conventions, but I'll explain using $0\in \Bbb N, 0 \notin \Bbb R_+$ and $N_n := \{k \in \Bbb N\mid k < n\}$ (other conventions do not change the answer, but modify the explanation).

1. The set of all infinite sequences with values in $\Bbb Z$ is indeed $\Bbb Z^{\Bbb N}$. For a fixed $n$, the set of all sequences of length $n$ into $\Bbb Z^{\Bbb N}$ is $\left(\Bbb Z^{\Bbb N}\right)^{N_n}$. So this set is $$S =\bigcup_{n=0}^\infty \left(\Bbb Z^{\Bbb N}\right)^{N_n}$$Since the sets being unioned are disjoint, we have $$|S| = \sum_{n=0}^\infty \left|\left(\Bbb Z^{\Bbb N}\right)^{N_n}\right| = \sum_{n=0}^\infty \left|\Bbb Z^{\Bbb N}\right|^{|N_n|} = \sum_{n=0}^\infty \mathfrak c^n$$Do you know what $\mathfrak c^n$ is for finite $n$?
2. Your analysis shows what happens for a fixed $n$. But the set description includes all finite sequences, regardless of length. This should not be too hard to fix.
3. $f(x) \mapsto e^xf(x)$ provides a bijection between $A$ and the set $A_1$ of all functions $f : \Bbb R_+ \to (0,1)$. And $\tan\left(\frac\pi2 x\right)$ provides a bijection between $(0,1)$ and $\Bbb R_+$.
4. $C$ is the set of sequences $h$ into $\{0,1\}$ such that for each $n, h$ takes on at least as many $0$ values as $1$ values at locations $\le n$. Consider an arbitrary sequence into $\{0,1\}$. We can build a new sequence by inserting an extra $0$ in front of each $1$ in the original sequence. This ensures that the resulting sequence will be in $C$. The process is also injective. You can convert the output sequence back into the original by removing any $0$ immediately preceding a $1$. So there is an injection from $\{0,1\}^{\Bbb N}$ into $C$.