I am trying to solve a limit problem for a sequence of functions. My professor defined the sequence as such
$\{u^n\}_{n=0}^{\infty} = \begin{cases} u^0 = 0\\ u^{n+1} = \mathcal{H}(u^n)\\ \end{cases}$
where $\mathcal{H}(u)$ is a linear mapping.
Is this just an alternative to the more common notation
$\{u_n\}_{n=0}^{\infty}$
The sequence describes different solutions to a linear second order inhomogenous ODE.
The common notation for sequences is to use parentheses instead of braces and subscripts instead of superscripts, that is $ (u_n)_{n = 0}^\infty \text{ or } (u_n)_{n \in \Bbb N}. $ The potential danger with the superscript notation is a possible confusion with an exponent. But maybe your professor has some specific reason to prefer the superscript notation in the context of linear second order nonhomogenous differential equations.