  A left-computable number $x$ is called regainingly approximable if there is a
computable increasing sequence $(x_n)_n$ of rational numbers converging to $x$
such that $x - x_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$; and it is
called nearly computable if there is such an $(x_n)_n$ such that for every
computable increasing function $s \colon \mathbb{N} \to \mathbb{N}$ the
sequence ${(x_{s(n+1)} - x_{s(n)})_n}$ converges computably to 0. In this
article we study the relationship between both concepts by constructing on the
one hand a non-computable number that is both regainingly approximable and
nearly computable, and on the other hand a left-computable number that is
nearly computable but not regainingly approximable; it then easily follows that
the two notions are incomparable with non-trivial intersection. With this
relationship clarified, we then hold the keys to answering an open question of
Merkle and Titov: they studied speedable numbers, that is, left-computable
numbers whose approximations can be sped up in a certain sense, and asked
whether, among the left-computable numbers, being Martin-L\"of random is
equivalent to being non-speedable. As we show that the concepts of speedable
and regainingly approximable numbers are equivalent within the nearly
computable numbers, our second construction provides a negative answer.
