User blog comment:QuasarBooster/Fibonacci/Lucas Sequence Extentions?/@comment-2033667-20150801003820

(by the way, the rest of the proof is $$\lim_{n \rightarrow \infty} \frac{F_n + nF_{n-1}}{F_{n-1} + (n-1)F_{n-2}} = \lim_{n \rightarrow \infty} \frac{F_n/(nF_{n-1}) + 1}{1/n + (n-1)F_{n-2}/(nF_{n-1})} = \frac{1}{\lim_{n \rightarrow \infty} \frac{(n-1)F_{n-2}}{nF_{n-1}}} = \frac{1}{1/\varphi} = \varphi.$$ I'm embarrassed to admit that LittlePeng had to help me with this)