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if the sequence [itex]a_n [/itex] satisfy that

[itex]a_{n+1}=a_n\left(1-c\frac{a_n}{1+a_n}\right)[/itex]

then [itex]a_n*c*n\rightarrow 1[/itex] for all positive [itex]c[/itex].

Like when [itex]c=1[/itex], then [itex]a_n*n\rightarrow 1[/itex],

but if [itex]c\neq 1[/itex], it's difficult to prove.