EDB — 0G0

Exercises

1. [0G0]Difficulty:*. Let \(I\) be a family of indices; let \(a_{i,j}:I× ℕ→ [0,∞]\) a generalised succession, such that \(j↦ a_{i,j}\) is weakly increasing for every fixed \(i\); prove that

   \[ ∑_{i∈ I} \lim _{j→∞} a_{i,j} = \lim _{j→∞} ∑_{i∈ I} a_{i,j}~ ~ . \]

   (This is a version of the well-known Monotone convergence theorem).

   Solution 1

   [0G2]↺↻

   [ [0G1]↺↻]