Let $$N=\{0,1,2,3,...\}$$ be the set of all non-negative integers and $A$ the set of all bijections from $N$ to itself.  Prove that   $i)$ $A$ is an infinite set.   $ii)$ There exist no bijection from $N$ to $A$
I guess it is too difficult a question to ask about the cardinality of $S_{\mathbb{N}}$ so I would like to ask whether it is countable or not.  I tried to prove it is uncountable somewhat mimicking the Cantor's diagonal argument but failed.
I am studying for an exam and I am having trouble with this practice question:  In this question, we consider finite bit strings that do not contain $00$. Examples of such bitstrings are $0101010101$ and $11110111$. For any integer $n\geq 2$, let $B_n$ be the number of bitstrings of length $n$ that do not contain $00$.   Determine $B_2$ and $B_3$. Prove that $B_n = B_{n-1} + B_{n-2}$ for each $n\geq 4$. For each $n\geq 2$, express $B_n$ in terms of a Fibonacci number.   Any help is greatly appreciated