  Many theorems of mathematics have the form that for a certain problem, e.g. a
differential equation or polynomial (in)equality, there exists a solution. The
sequential version then states that for a sequence of problems, there is a
sequence of solutions. The original and sequential theorem can often be proved
via the same (or similar) proof and often have the same (or similar) logical
properties, esp. if everything is formulated in the language of second-order
arithmetic. In this paper, we identify basic theorems of third-order
arithmetic, e.g. concerning semi-continuous functions, such that the sequential
versions have very different logical properties. In particular, depending on
the constructive status of the original theorem, very different and independent
choice principles are needed. Despite these differences, the associated Reverse
Mathematics, working in Kohlenbach's higher-order framework, is rather elegant
and is still based at the core on weak K\"onig's lemma.
