The third positive element in the greedy $B_h$-set

For $h\ge 1$, a $B_h$-set is a set of integers such that every integer $n$ has at most one representation in the form $n=a_{i_1}+\cdots +a_{i_h}$, where $a_{i_j}\in A$ for all $j=1,\dots ,h$ and $a_{i_1}\le \dots \le a_{i_h}$. The greedy $B_h$-set is the infinite set of nonnegative integers $\{a_0(h),a_1(h),a_2(h),\dots \}$ constructed as follows: If $a_0(h)=0$ and $\{a_0(h),a_1(h),a_2(h),\dots ,a_k(h)\}$ is a $B_h$-set, then $a_{k+1}(h)$ is the least positive integer such that $\{a_0(h),a_1(h),a_2(h),\dots ,a_k(h),a_{k+1}(h)\}$ is a $B_h$ set. One has $a_1(h)=1$ and $a_2(h)=h+1$ for all $h$. Elementary proofs are given that $a_3(h)=h^2+h+1$ for all $h\ge 1$ and that $a_k(h)\le {\sum }_{i=0}^{k-1}{h}^i$ for all $h\ge 1$ and $k\ge 1$.