Wow!

$\lim _{ n\rightarrow \infty }{ \frac { { a }_{ n-1 } }{ { a }_{ n } } } \\ \\ { F }_{ n }=\frac { { a }_{ n+1 } }{ { a }_{ n } } =\frac { { a }_{ n }+{ a }_{ n-1 } }{ { a }_{ n } } \\ \\ =1+\frac { { a }_{ n-1 } }{ { a }_{ n } } \\ =1+\frac { 1 }{ { F }_{ n-1 } } \\ \\ \left| { F }_{ n }-\phi \right| =\left| 1+\frac { 1 }{ { F }_{ n-1 } } -1-\frac { 1 }{ \phi } \right| ,\quad by\quad the\quad definition\quad of\quad \phi \quad we\quad have\quad \phi =1+\frac { 1 }{ \phi } \\ \\ \\ \left| { F }_{ n }-\phi \right| =\left| \frac { 1 }{ { F }_{ n-1 } } -\frac { 1 }{ \phi } \right| ,then\quad we\quad add\quad the\quad fractions\\ \\ =\left| \frac { { F }_{ n-1 }+\phi }{ { F }_{ n-1 }\phi } \right| \\ \\ =\frac { 1 }{ \phi } \left| 1+\frac { \phi }{ { F }_{ n-1 } } \right| \\ \\ \le \frac { 1 }{ \phi } \left| { F }_{ n-1 }-\phi \right| \\ \\ \le (\frac { 1 }{ \phi } )^{ n-1 }\left| { F }_{ 1 }-\phi \right| \\$

as n tends to $\infty ,(\frac { 1 }{ \phi } )^{ n-1 }$ tends to 0 so ${ F }_{ n }$ tends to

$\phi$ !