An interesting result

Define a recursive series x_n with x_0=0
x_n+1 = x_n + f(n)/(10^n)
let f(n) be any function from rationals to rationals, then the limit as n goes to infinity of x_n is always rational
an example: if we let f(n) be the nth fibonacci number, the limit is 1/89
i choose this specific example to show that even functions that are a bit strange in their definitions satisfy this
also note that f(n) has to grow slower than 10^n or else the limit goes to infinity
this is all totally conjecture and unproven. if you can provide a counter example i would be quite happy