Suppose $f\_1,f\_2,\ldots $ is a sequence of convex functions that converges to a continuous convex $f$. Let $x\_1^\*,x\_2^\*$ be their respective (not necessarily unique) minima, and let y be a minima of $f$ (once again need not be unique). Can we prove that there exists a version of $x\_1^\*,x\_2^\*,\ldots$ such that $x\_n^\*\rightarrow y$ ? 



Does the minima of a sequence of convex convergent functions converge?
No; here's a counterexample: let $f = 0$ and consider the minimizer $y = 0.$ Then you can construct convex functions which converge to $0$ pointwise but whose minima are always moving away from $y =0,$ e.g. $f\_n(x) = (x - n)^2/n^n.$ 


