Does this follow from prop II.8.1.5 (p. 165) in
http://wolfweb.unr.edu/homepage/bruceb/Cycr.pdf
?
Applied once for each (a, alpha, epsilon > 0) with X := the closed real
interval [0, ||c_alpha||], and f(x) := sqrt(x) to obtain:
There exists delta > 0 with
||a((c_alpha)^(1/2)) - ((c_alpha)^(1/2))a|| < ||a||(epsilon)
whenever
||a(c_alpha) - (c_alpha)a|| < ||a||(delta)