How to prove this Stirling related equation Here is what I need to prove, but have no idea were to start. I know there is some connection with the Stirling theorem.
$$
\sum_{i=0}^{d}\binom{m}{i} \leq \left ( \frac{em}{d} \right )^{d}
$$
for $$ m \geq d$$
I tried to open the sum and start from the right side of the equation but I achieved nothing. I think there is also some connection to Sauer Shelah Lemma.
Thanks in advance.

I think it is the Sauer Shelah Lemma isn't it?
There are proofs at the link below; one of which uses a "shifting technique" and the other induction on $m+d$. 
http://www.cse.buffalo.edu/~hungngo/classes/2011/Fall-694/lectures/vcd.pdf
See references therein to trace the history.
