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\lhead{\bf Advanced Calculus (MAA 4211)}
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\rhead{Due Wednesday 10/17}
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\begin{enumerate}

\Problem By negating the definition of convergence of a sequence, state what it means for the sequence $(a_n)$ from the metric space $(X,d)$ to \emph{not converge}.

\Problem \emph{Using only the definition of convergence}, show that the sequence $(a_n)$ given by $a_n=(-1)^n$ does not converge in $(\mathbb{R},d_1)$.

\Problem Suppose $(a_n)$ is a sequence from $\mathbb R$. Show that if $(a_n)$ converges to $L$ then the sequence (of Cesaro means) $(s_n)$ defined by
\[
	s_n = \frac{1}{n+1} \sum_{j=0}^n a_j
\]
also converges to $L$. Does the converse also hold?

\end{enumerate}

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