EDB — 0D0

Exercises

1. [0D0]Prerequisites:[0CX]↺↻,[0CS]↺↻.Let again \(a_{n,m}\) be a real valued sequence with two indices \(n,m∈ℕ\); suppose that, for every \(n\), the limit \(\lim _{m→ ∞} a_{n,m}=b_ n\) exists, is finite and is uniform in \(n\); suppose that the limit \(\lim _ n b_ n\) exists and is finite. Can it be concluded that the limits \(\lim _{n→ ∞}a_{n,m}\) exist for each fixed \(m\)? Can we write an equality as in eqn. [(7.3)]↺↻ in which, however, on the RHS we use the upper or lower limits of \(a_{n,m}\) for \(n→ ∞\), instead of the limits \(\lim _{n→ ∞}a_{n,m}\)?

   Solution 1

   [0D1]↺↻