Showing that if the limit of norms converges, then the sequence converges

Showing that if the limit of norms converges, then the sequence converges

Let H be a hilbert space, and C subset H a convex set. Let x_n_nin mathbbN be a sequence in C with lim_nto inftyx_n inf_xin Cx. Show x_n converges in H. So far I have: Let P_C0 x in C: x inf_x in Cx be the projection of 0 onto C. If we consider barC the closure of C, then there is a theorem that tells us since barC is closed and convex, there is exactly one y in barC with yP_barC0, i.e., y inf_xin barCx. Since the infimum of a sets closure equals the infimum of the set, we also have y inf_xin C x. Now, I need to show that since the norms converge to the norm of a unique element y, the sequence itself must converge to this element. This makes sense intuitively, but I cant make it rigorous Any hints would be appreciated.

x_n-x_m2x_nx_m22x_n22x_m2. Denoting inf_xin C x by A we get limsup x_n-x_m2 leq 2A2A-liminf 4frac x_nx_m 22. Note that frac x_nx_m 2 in C so limsup x_n-x_m2 leq 2A22A2-4A20. x_n is Cauchy, hence convergent.

This is actually true in the more general setting of uniformly convex Banach spaces. As x_n converges to y inf_xin C x, we can take some radius r y and note that x_n_nin mathbbN is eventually in overlineCcap B_r0, where B_r0 is the closed ball of radius r centered at 0. The set overlineCcap B_r0 is the intersection of a weakly closed set and a weakly compact set and is therefore weakly compact. For a subsequence x_n_k_kin mathbbN, we let its weakly convergent subsequence be x_n_k_j_jin mathbbN which exists by the Eberlein-mulian theorem, and we let x_n_k_jrightharpoonup xin overlineCcap B_r0. By the weak lower semicontinuity of the norm, we have xleq liminf_jto infty x_n_k_j y which is of course impossible unless x y by our definition of y as the unique element of overlineC with minimal norm. Therefore, as every subsequence has a further subsequence that converges weakly to y, we have x_nrightharpoonup y. Then, weak convergence and norm convergence together imply strong convergence in uniformly convex Banach spaces, so we have x_nto y.