Approximation to Golden Rectangle using Fibonacci Squares

Theorem
An approximation to a golden rectangle can be obtained by placing adjacent to one another squares with side lengths corresponding to consecutive Fibonacci numbers in the following manner:


 * FibonacciRectangle.png

It can also be noted, as from Sequence of Golden Rectangles, that an equiangular spiral can be approximated by constructing quarter circles as indicated.

Proof 2
Let the last two squares to be added have side lengths of $F_{n - 1}$ and $F_n$.

Then from the method of construction, the sides of the rectangle generated will be $F_n$ and $F_{n + 1}$.

From Continued Fraction Expansion of Golden Mean it follows that the limit of the ratio of the side lengths of the rectangle, as $n$ tends to infinity, is the golden section $\phi$.

Hence the result.