We show that if a is an even integer then for every ξ ∈ R the smallest limit point of the sequence ||ξan||∞n=1 does not exceed |a|/(2|a| + 2) and this bound is best possible in the sense that for some ξ this constant cannot be improved. Similar (best possible) bound is also obtained for the smallest limit point of the sequence ||ξxn||∞n=1, where (xn)∞n=1 satisfies the second order linear recurrence xn = axn−1 + bxn−2 with a, b ∈ N satisfying a > b. For the Fibonacci sequence (Fn)∞n=1 our result implies that supξ∈R lim infn→∞ ||ξFn|| = 1/5, and e.g., in case when a > 3 is an odd integer, b = 1 and θ := a/2 + p a 2/4 + 1 it shows that supξ∈R lim infn→∞ ||ξθn|| = (a − 1)/2a.
Bounding squares in second order recurrence sequences