Limit behaviour of constant distance boundaries of Jordan curves

For a Jordan curve $ \Gamma $ in the complex plane, its constant distance boundary $ \Gamma_ \lambda $ is an inflated version of $ \Gamma $. A flatness condition, $ (1/2, r_0) $-chordal property, guarantees that $ \Gamma_ \lambda $ is a Jordan curve when $ \lambda $ is not too large. We prove that $ \Gamma_ \lambda $ converges to $ \Gamma $, as $ \lambda $ approaching to $ 0 $, in the sense of Hausdorff distance if $ \Gamma $ has the $ (1/2, r_0) $-chordal property.