  We prove the enveloping property of the known divergent asymptotic expansions
of the large real zeros of the cylinder and Airy functions, and thereby
answering in the affirmative two conjectures posed by Elbert and Laforgia and
by Fabijonas and Olver, respectively. The essence of the proof is the
construction of analytic functions that return the zeros when evaluated along
certain discrete sets of real numbers. By manipulating contour integrals of
these functions, we derive the asymptotic expansions of the large zeros
truncated after a finite number of terms plus remainders that can be estimated
efficiently. The conjectures are then deduced as corollaries of these
estimates. An analogous result for the associated phase function is also
discussed.
