This is my approach:
$|{\hat \gamma}(h)| = \frac{1}{n}\sum_{t=1}^{n-|h|}(X_{t}-{\bar X})(X_{t+|h|}-{\bar X}) \le \frac{1}{n}\sum_{t=1}^{n}(X_{t}-{\bar X})(X_{t+|h|}-{\bar X}) = \frac{1}{n}\sum_{t=1}^{n}(X_{t}-{\bar X})(X_{t}-{\bar X}) + \frac{1}{n}\sum_{t=1}^{n}(X_{t + |h|}-X_{t})(X_{t}-{\bar X}) = {\hat \gamma}(0) + \frac{1}{n}\sum_{t=1}^{n}(X_{t + |h|}-X_{t})(X_{t}-{\bar X})$
Can I proceed further? Or should I approach in a different way?