EDB — 0B9

Exercises

1. [0B9] Prerequisites:[0B7]↺↻,[0B6]↺↻.(Solved on 2022-11-24) Let \(a_ n,b_ n\) be real sequences, for \(n∈ I\), show that

   \[ \sup _{n,m∈ I}(a_ n+b_ m) = (\sup _{n∈ I} a_ n) + (\sup _{n∈ I} b_ n)~ ~ , \]

   but

   \[ \sup _{n∈ I}(a_ n+b_ n)≤ (\sup _{n∈ I} a_ n) + (\sup _{n∈ I} b_ n)~ ~ ; \]

   find a case where inequality is strict.

   Solution 1

   [0BB]↺↻