EDB — 0CP

Exercises

1. [0CP] Prerequisites:[20M]↺↻.
   We have that \(\sup _{n≥ N} a_ n=𝜎∈ \overlineℝ\) if and only if

   \begin{eqnarray} & & ∀ n≥ N, a_ n≤ 𝜎 \label{eq:sup_ con_ ge_ L} \quad \text{e}\\ & & ∀ L{\lt} 𝜎, ∃ n≥ N, a_ n {\gt}L \label{eq:sup_ con_ le_ L} \end{eqnarray}

   (note that if \(𝜎=∞\) the first is trivially true, while if \(𝜎=-∞\) the latter is true because there are no \(L\)).
   Solution. [0CQ]↺↻\(\square \)