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Theorem:

Let $a_n$ be a real sequence convergent to $a \in \mathbb{R}$. Let $c_{k,n}$ (where $1\le k \le n$) be a sequence such that:

$$\quad \forall k \lim_{n \to \infty}c_{k,n} = 0$$ $$\quad \lim_{n \to > \infty} \sum_{k=1}^n c_{k,n} = 1$$ $$\quad \exists M>0 : \forall n\ \ > \sum_{k=1}^n |c_{k,n}| \le M$$

Then $\lim_{n \to \infty}s_n =a$, where $$s_n \equiv \sum_{k=1}^n c_{k,n} \cdot a_k.$$

The author of my textbook begins by observing that if $a_n$ is a constant sequence then $$s_n=a\sum_{k=1}^n c_{k,n}$$ implying that $\lim_{n\to \infty}s_n=a.$ He then remarks that it is enough to consider the case when the sequence is equal to zero. I don't understand the reasoning behind this argument and would, therefore, be grateful if someone could explain this step in the proof. Here is the complete proof by the way. enter image description here

Andrés E. Caicedo
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Student
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1 Answers1

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If $(a_n)_n$ converges to $a$ then the sequence $(a_n-a)_n$ converges to $0$. If we assume that the theorem holds for sequences converging to zero, we know that for $$ s_n = \sum_{k=1}^n c_{k,n} (a_k-a) $$ we have $\lim_{n \to \infty} s_n =0$. But you have also just proved that for $$ s_n'= \sum_{k=1}^n c_{k,n} a $$ we have $\lim_{n \to \infty} s_n' = a$. Note that if we define $$ S_n = \sum_{k=1}^n c_{k,n} a_k = s_n +s_n', $$ then we find $\lim_{n \to \infty} S_n = 0+a = a$. So the theorem holds in general.

Demophilus
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