I have to prove that, $\forall n\geq 1$, $\forall 1\leq k\leq n$, $$ \frac{k^{n}}{k^{k}}\leq S(n,k)\leq \frac{k^{n}}{k!} $$
The second inequality, is quite obvious for me, as $k!\cdot S(n,k)$ is the number of surjective applications from a set of $n$ elements to a set of $k$ elements ($k^{n}$ is the total number of applications from a set of $n$ elements to a set of $k$ elements), so, $k!S(n,k)\leq k^{n}$ $\Rightarrow$ $S(n,k)\leq \frac{k^{n}}{k!} $.
Nevertheless I don't really know how to approach the first inequality... Probably I could relate it again with the number of applications,... I don't really know.
A hint / some help would be so useful! Thanks in advanced!
Hint: So, what if you have a set partition where you know $1,2,\cdots ,k$ are in different parts. In how many ways can you assigned the other $n-k$ elements?
Are these all possible partitions of $[n]$ into $k$ blocks?