bounding the height of stack when checking acceptance pushdown automaton

Let $A$ be a nondeterministic PDA (with empty stack acceptance). I am looking for a reference for a statement of the following form.

There exists a constant $c$, computable from $A$, such that:
if $w$ is accepted by $A$ then there is an accepting computation of $w$ on $A$ with stack height at most $c|w|$.

From this it would follow that we can decide whether $A$ accepts $w$ without going via equivalent context-free grammars.