In [Does the Busy Beaver function grow faster than the Tree function?][1], the informal proof hinges on non-computable functions such as Busy Beaver growing faster than computable ones such as TREE. *Is this necessarily given?* For example, a trivially non-computable function which grows slower than TREE would be the sum of reciprocals of Busy Beaver: $$\sum_{i=1}^n\frac{1}{BB(n)}$$

  [1]: https://www.quora.com/Does-the-Busy-Beaver-function-grow-faster-than-the-Tree-function