# 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.