Some results on strong limit theorems for (LB)-space-valued random variables

Let E be a strict (LB)-space, i.e., a strict inductive limit of separable Banach spaces E1 [subset of] E2 [subset of] ... One of the results proved in this is the following. Let Xn, n [greater-or-equal, slanted] 1 be a sequence of independent identically distributed (i.i.d.) random variables taking values in E. If the Strong law of large numbers holds for this sequence, i.e., (1/m)[summation operator]i = 1m Xi, m [greater-or-equal, slanted] 1 converges almost surely, then there exists n [greater-or-equal, slanted] 1 such that each Xi takes values in En almost surely.