I wonder if what I show below is allowed and why.
Suppose you have an interval $ (0, T) $. For each $ n $ you divide the
interval in $ 2 ^ n $ n parts. In each interval, you have a function
$ o ( Delta t) $. We have the summation$ sum limits_ {i = 1} ^ {2 ^ n} o ( Delta t) = 2 ^ no ( Delta t) = T frac {o ( Delta
t)} { frac {T} {2 ^ n}} = T frac {o ( Delta t)} { Delta t} $.So, it goes to zero as n goes to infinity.
I saw this argumentation in a book. Is it permissible or do we need more constraints on the $ o ( Delta t) $ functions? The problems I have are:

Let n, the $ o ( Delta t) $ the functions may be different, so can we just say that they are $ 2 ^ n $ once a $ o ( Delta t) $ a function?

When $ n $ increases, we get new $ o ( Delta t) $functions. Does this affect the outcome? For the limit to reach zero as $ n $ increases we need to be the same $ o ( Delta t) $a function?
So, is the result valid or are there counterexamples where they are not? Do we need more conditions on the functions for this to work?