Consider the differential equation z ˙ = ε f ( z , t , ε ) \dot z = \varepsilon f(z,\,t,\,\varepsilon ) , where f f is T T periodic in t t and ε > 0 \varepsilon > 0 is a small parameter, and the averaged equation z ˙ = f ¯ ( z ) := ( 1 / T ) ∫ 0 T f ( z , t , 0 ) d t \dot z = \overline f (z): = (1/T)\,\int _0^T {\,f(z,\,t,\,0)\,dt} . Suppose the averaged equation has a hyperbolic equilibrium at z = 0 z = 0 with stable manifold W ¯ \overline W . Let β ε ( t ) {\beta _\varepsilon }(t) denote the hyperbolic T T -periodic solution of z ˙ = ε f ( z , t , ε ) \dot z = \varepsilon f(z,\,t,\,\varepsilon ) near z ≡ 0 z \equiv 0 . We prove a result about smooth convergence of the stable manifold of β ε ( t ) {\beta _\varepsilon }(t) to W ¯ × R \overline W \times {\mathbf {R}} as ε → 0 \varepsilon \to 0 . The proof uses ideas of Vanderbauwhede and van Gils about contractions on a scale of Banach spaces.