Prove consistency of maximum of two consistent heuristic functions? Suppose we have two heuristic functions $h_1$ and $h_2$ which are both consistent, that is, 

$$h_i(n) \leq c(n,a,n') + h_i(n')\qquad \ i\in\{1,2\}\,,$$

where $c(n,a,n')$ denotes cost of reaching the successor node $n'$ with action $a$.
Can we conclude that the $h_3:=\max(h_1,h_2)$ is also consistent?
Proof (Show consistency property of $h_3$):

$$
h_3(n) = \max(h_1(n), h_2(n)) \\ 
\leq max(h_1(n')+c(n,a,n'), \ h_2(n')+c(n,a,n')) \\ 
\leq \max(h_1(n'), \ h_2(n')) + c(n,a,n') = h_3(n') + c(n,a,n')
$$