http://math.harvard.edu/~ctm/home/text/books/royden-fitzpatrick/royden-fitzpatrick.pdf
In the book of Royden, the completeness of $ L^{p} $ spaces has been done using what he calls "rapidly Cauchy" sequences.
A sequence $ f_{n} \in X $, where $ X $ is a normed linear is said to be rapidly Cauchy if there is a sequence of positive reals $ \epsilon_{k} $ such that $ \sum_{ k = 1 } ^ { \infty} \epsilon_{k} $ is convergent and
$$ || f_{k+1} - f_{k} || < \epsilon_{k} ^{2} , $$
for all $ k \in \mathbb{N} $.
However, I don't understand why do we need to square the $ \epsilon_{k} $. The proofs all work fine even if I just put $ \epsilon_{k} $ as an upper bound on the $ || f_{k+1} - f_{k} || $ norm. As far as I see, all propositions and theorems up to the Reisz-Fischer Theorem remain valid.
Question. What purpose does squaring the $ \epsilon_{k} $ term serve in the chapter?