Upper bound for partial sum of binomial coefficients: $\sum\limits_{i=0}^k \binom{n}{i} \le (n+1)^k$
Define $E:=\{1,...,n\}$ to be a set of cardinal $n$ :
$$X:=\{A\subseteq E\mid |A|\leq k\}$$
$$Y:=\{f:\{1,...k\}\rightarrow E\cup\{0\}\}$$
Now :
$$\psi : X\rightarrow Y $$
$$A:=\{a_1<...<a_l\}\mapsto f_A $$
Remarking that $|X|\leq |Y|$ is exactly the inequality you want, you just have to find a way to define $f_A$ so that the function $\psi$ is one-to-one.
Hint : Use the fact that any $A\in X$ can be written as $A=\{a_1<...<a_l\}$ with $0\leq l\leq k$.