Prove that if two normed spaces are Lipschitz equivalent, then one if complete iff the other is.
Let $ (V_1, \Vert\cdot\Vert_1) $ and $ (V_2, \Vert\cdot\Vert_2) $ be Lipschitz equivalent normed vector spaces. Then there exists $f : V_1 \to V_2 $, and constants $h, k > 0 $, such that $ h\Vert f(x) - f(y)\Vert_2 \leq \Vert x-y\Vert_1 \leq k\Vert f(x) - f(y)\Vert_2 $ for all $ x,y, \in V_1 $. Suppose $(V_2, \Vert\cdot\Vert_2) $ is complete.
Clearly everything is symmetrical, so we only really need to prove this in one direction. I can see that if $ (x_n) $ is a Cauchy sequence in $V_1$, then $(f(x_n))$ is Cauchy in $ V_2 $. I can also see that $ f $ is uniformly continuous. How can I turn this into a proof?