  Often in the analysis of first-order methods, assuming the existence of a
quadratic growth bound (a generalization of strong convexity) facilitates much
stronger convergence analysis. Hence the analysis is done twice, once for the
general case and once for the growth bounded case. We give a meta-theorem for
deriving general convergence rates from those assuming a growth lower bound.
Applying this simple but conceptually powerful tool to the proximal point
method, the subgradient method, and the bundle method immediately recovers
their known convergence rates for general convex optimization problems from
their specialized rates. Future works studying first-order methods can assume
growth bounds for the sake of analysis without hampering the generality of the
results. Our results can be applied to lift any rate based on a H\"older growth
bound. As a consequence, guarantees for minimizing sharp functions imply
guarantees for both general functions and those satisfying quadratic growth.
