A little help with a proof?

I'm not very good at writing good proofs of stuff just yet, so I thought I'd ask my dear friends /r/math about it.

I begun by proving (I think) that [; \lim_{n \to \infty }\frac{F_{n+1}}{F_n} = \varphi ;].

We begin by defining [; \varphi ;] as the positive solution to [; \frac{1}{x} +1 = x ;]

A bunch of simple algebra: [; \lim_{n \to \infty }\frac{F_{n+1}}{F_n} = \lim_{n \to \infty }\frac{F_n + F_{n-1}}{F_n} ;] [;= \lim_{n \to \infty}\left ( 1 + \frac{F_{n}}{F_{n-1}} \right ) =;][; \lim_{n \to \infty} 1 + \left (\frac{F_{n-1}}{F_{n}} \right )^{-1}} ;]

Observe that, as we are taking a limit to infinity, [;\lim_{n \to \infty}\frac{F_{n+1+k}}{F_{n+k}} = \lim_{n \to \infty}\frac{F_{n+1}}{F_{n}} ;] for any integer value of k, so: [;\lim_{n \to \infty} 1 + \left (\frac{F_{n}}{F_{n-1}} \right )^{-1} = \lim_{n \to \infty} 1 + \left (\frac{F_{n+1}}{F_{n}} \right )^{-1} ;]

Thus [; \lim_{n \to \infty} \frac{F_{n+1}}{F_n} ;] is a solution to our equation and must clearly be positive, and is therefore equal to [; \varphi ;]

Is this a valid proof?

I'll post the second question in a comment to this, to not make it completely tl;dr.

EDIT: Apparently my LaTeX only works in the RES live preview?!