Prove that sqrt(1+sqrt(1+sqrt(1+...))) =
1 + 1/(1+1/(1+1/...)) = Phi

(What a wonderful world...!)

sqrt(1+sqrt(1+sqrt(1+...))) = X
1+sqrt(1+sqrt(1+...))) = X^2
1 + X = X^2
X^2 - X - 1 = 0
X = (1 + sqrt(1+4))/2 or (1 - sqrt(1+4))/2

1 + 1/(1+1/(1+1/...)) = X
1 + 1/X = X
X + 1 = X^2
X^2 - X - 1 = 0
X = (1 + sqrt(1+4))/2 or (1 - sqrt(1+4))/2

Almost perfect! But there is an important and rather hard detail to be proved. How do we know that sqrt(1+sqrt(1+sqrt(1+...))) or 1 + 1/(1+1/(1+1/...)) are not infinity? And also how do we know that they are well-defined??? For example: 1-1+1-1+... or
2(2/(2(2/....))) are not well defined!!! You should treat these quantities as limits and not as numbers...