The Number of Terms in Greedy Fibonacci–Lucas Representations

We prove Kimberling's conjecture on greedy representations by Fibonacci and Lucas numbers: the ratio of the number of Zeckendorf summands of $1,\dots,n$ to the number of terms of their greedy representations tends to a limit in $(1.2,1.4)$, and the limit is $(5-\sqrt5)/2$. We also prove the conjectures of Greathouse and Kurkov on the number of terms of the greedy representation.