EDB — 0CX

Exercises

1. [0CX] Difficulty:*.

   Let \(a_{n,m}\) be a real valued sequence ¹ with two indexes \(n,m∈ℕ\). Suppose that

   • for every \(m\) the limit \(\lim _{n→ ∞} a_{n,m}\) exists, and that

   • the limit \(\lim _{m→ ∞} a_{n,m}=b_ n\) exists uniformly in \(n\) and is finite, i.e.

     \[ ∀ \varepsilon {\gt}0 ,~ ∃ m∈ℕ~ ∀ n∈ℕ ,~ ∀ h≥ m ~ ~ | a_{n,h}-b_ n|{\lt}\varepsilon ~ ~ . \]

   then

   \begin{equation} \lim _{n→ ∞} \lim _{m→ ∞} a_{n,m}= \lim _{m→ ∞} \lim _{n→ ∞} a_{n,m}\label{eq:limlimlimlim} \end{equation}

   3

   in the sense that if one of the two limits exists (possibly infinite), then the other also exists, and they are equal.

   Find a simple example where the two limits in 3 are infinite.

   Find an example where \(\lim _{m→ ∞} a_{n,m}=b_ n\) but the limit is not uniform and the previous equality 3 does not apply.

   Solution 1

   [0CZ]↺↻