Argue that the sequence \(1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,...\) does not converge to zero. For what values of \(\epsilon > 0\) does there exist a response \(N\)? For which values of \(\epsilon > 0\) is there no suitable response.
Definitions of sequences and convergence: here, Definitions of subsequences: here.
Solution
Let \(\{a_n\}\) be the sequence above. A response means an \(N\) value that satisfies the definition of convergence. Observe that when \(\epsilon > 1\), then
$$
\begin{align*}
\left| a_n - 0 \right| = \left| a_n \right| &< \epsilon < 1
\end{align*}
$$
Since \(a_n\) is either \(1\) or \(0\), then this inequality holds for all \(n\) so it suffices to set \(N = 1\). However, suppose that \(\epsilon \leq 1\).
$$
\begin{align*}
\left| a_n \right| &< \epsilon \leq 1
\end{align*}
$$
Since the sequence contains infinitely many terms equal to \(1\), no matter how large \(N\) is, there will always exist some \(n \geq N\) such that \(a_n = 1\) and therefore, violating the condition that \(|a_n| < \epsilon\).
References