EDB — 0B6

Exercises

1. [0B6] (Solved on 2022-11-24) Let \(a_ n\) be a real-valued sequence, for \(n∈ I\) a set of indexes; let \(r{\gt}0,t∈ℝ,𝜌{\lt}0\); show that

   \[ \sup _{n∈ I}(a_ n+t)=t+\sup _{n∈ I}a_ n~ ~ ,~ ~ \sup _{n∈ I}(r a_ n)=r \sup _{n∈ I}a_ n~ ~ ,~ ~ \sup _{n∈ I}(𝜌 a_ n)= 𝜌 \inf _{n∈ I}a_ n~ ~ . \]
   Solution 1

   [22W]↺↻