Given $\{a_i\}_{i=1}^{\infty}$ as a sequence of real numbers, define $\{c_j\}_{j=1}^{\infty}$ Such that for each $j \in \{1,2,3,...\},$ We know that $$c_j = \frac{\sum_{i=1}^{j} a_i}{j}.$$ I want to show that if $\lim_{j \rightarrow \infty} a_j = q,$ then $\lim_{j \rightarrow \infty} c_j = q.$ That being said, I am not sure about the tools I have available to go about solving this problem. I do know that $\exists N \in \mathbb{N}$ such that for all $\epsilon > 0,$ $|a_j-q| < \epsilon$ for $j > N.$ I would like to go about using this knowledge to bound $|m_j - q|$ with a function of $\epsilon.$
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