  Often in the analysis of first-order methods for both smooth and nonsmooth
optimization, assuming the existence of a growth/error bound or a KL condition
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
meta-theorems for deriving general convergence rates from those assuming a
growth lower bound. Applying this simple but conceptually powerful tool to the
proximal point method, subgradient method, bundle method, gradient descent and
universal accelerated method immediately recovers their known convergence rates
for general convex optimization problems from their specialized rates. Our
results apply to lift any rate based on H\"older continuity of the objective's
gradient and H\"older growth bounds to apply to any problem with a weaker
growth bound or when no growth bound is assumed.
