Exercise 2.6.5

Question

Consider the following (invented) definition: A sequence $\left(s_{n}\right)$ is pseudo-Cauchy if, for all $\epsilon>0$, there exists an $N$ such that if $n \geq N$, then $\left|s_{n+1}-s_{n}\right|<\epsilon$

Decide which one of the following two propositions is actually true. Supply a proof for the valid statement and a counterexample for the other.
  \begin{enumerate}[label=(\roman*)] 
  \item Pseudo-Cauchy sequences are bounded.
  \item  If $\left(x_{n}\right)$ and $\left(y_{n}\right)$ are pseudo-Cauchy, then $\left(x_{n}+y_{n}\right)$ is pseudo-Cauchy as well.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\roman*)] 
  \item False, consider $s_n = \log n$. clearly $|s_{n+1} - s_n|$ can be made arbitrarily small but $s_n$ is unbounded.
  \item True, as $|(x_{n+1} + y_{n+1}) - (x_n + y_n)| \le |x_{n+1} - x_n| + |y_{n+1} - y_n| < \epsilon/2 + \epsilon/2 = \epsilon$.
   \end{enumerate}

Exercise 2.6.5

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Exercise 2.6.5

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