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The theorem states that if you have an infinite matrix of non-negative real numbers , such that the rows are weakly increasing and each is bounded , where the bounds are summable < then, for each column, the non decreasing column sums , are bounded hence convergent, and the limit of the column sums is equal to the sum of the "limit column ...
Firstly, we will acknowledge that a sequence () (in or ) has a convergent subsequence if and only if there exists a countable set where is the index set of the sequence such that () converges. Let ( x n ) {\displaystyle (x_{n})} be any bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} and denote its index set by I {\displaystyle I} .
A sequence ,, … of real-valued ... for all bounded, Lipschitz functions; () ... variables such that the almost surely convergent sequences are exactly the ...
The set of all convergent sequences is a vector subspace of called the space of convergent sequences. Since every convergent sequence is bounded, c {\displaystyle c} is a linear subspace of ℓ ∞ . {\displaystyle \ell ^{\infty }.}
Every bounded sequence in has a convergent subsequence, by the Bolzano–Weierstrass theorem. If these subsequences all have the same limit, then the original sequence also converges to that limit. If these subsequences all have the same limit, then the original sequence also converges to that limit.
(Monotone convergence theorem) If is bounded and monotonic for all greater than some , then it is convergent. A sequence is convergent if and only if every subsequence is convergent. If every subsequence of a sequence has its own subsequence which converges to the same point, then the original sequence converges to that point.
So let f be such arbitrary bounded continuous function. Now consider the function of a single variable g(x) := f(x, c). This will obviously be also bounded and continuous, and therefore by the portmanteau lemma for sequence {X n} converging in distribution to X, we will have that E[g(X n)] → E[g(X)].
Every uniformly convergent sequence of bounded functions is uniformly bounded.; The family of functions () = defined for real with traveling through the integers, is uniformly bounded by 1.