HMMT 2012 T8

For integer $n, m \geq 1$, let $A(n, m)$ denote the number of functions $f:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, m\}$ such that $f(j)-f(i) \leq j-i$ for all $1 \leq i<j \leq n$, and let $B(n, m)$ denote the number of functions $g:\{0,1, \ldots, 2 n+m\} \rightarrow\{0,1, \ldots, m\}$ such that $g(0)=0, g(2 n+m)=m$, and $|g(i)-g(i-1)|=1$ for all $1 \leq i \leq 2 n+m$. Prove that $A(n, m)=B(n, m)$.