Number of partitions of a finite set

We denote by $B_n$ the $n$-th Bell number, that is, the number of partitions of a set with $n$ elements. We set $B_0 = 1$.

1. Show that $B_{n+1} = \displaystyle\sum_{k=0}^{n} \binom{n}{k} B_k$.

2. Show that $B_n\leqslant n!$ for all $n\geqslant0$. Conclude that the exponential generating function $S(x) := \sum_{n\geqslant0} \dfrac{B_n}{n!} x^n$ is defined in a neighborhood of $0$.

3. Show that $S(x)=\displaystyle e^{e^x-1}$ for all $x$ close to $0$.

4. Conclude that $B_n=\displaystyle\frac{1}{e}\sum_{k=0}^{+\infty}\frac{k^n}{k!}$.

5. Deduce the number of equivalence relations that can be considered on a set with $n$ elements.