Topology
True or False? These curves converge pointwise to the unit circle.
Details.
There is an open circle at the point \((2,0),\) and a linear portion with slope \(\frac{-1}{n+1}\) at Stage \(n\) that continues until it meets the unit circle, and then follows the circle around counterclockwise to \((1,0).\)
Let \(f_n(\theta)\) be the point where a ray making an angle of \(\theta\) with the positive \(x-\)axis intersects the Stage \(n\) curve \(f_n,\) and let \(f(\theta)\) be the point where the same ray intersects the unit circle. A picture of \(f_2(\theta)\) for an angle \(\theta\) is shown below.
True or False? For each \(\theta\) in the interval \([0, 2\pi),\)
\[\lim_{n \to \infty} f_n(\theta) = f(\theta).\]
True or False? The derivatives of the curves converge to the derivative of the circle.
Details. There is an open circle at the point \((2,0),\) and a linear portion with slope \(\frac{-1}{n+1}\) at Stage \(n\) that continues until it meets the unit circle, and then follows the circle around counterclockwise to \((1,0).\)
Let \(d_n(\theta)\) be the derivative \(\frac{dy}{dx}\) of the Stage \(n\) curve at the point where it intersects a ray making an angle of \(\theta\) with the positive \(x-\)axis, and let \(d(\theta)\) be the derivative \(\frac{dy}{dx}\) of the unit circle at the point where the same ray intersects the unit circle.
True or False? For each \(\theta\) in the interval \((0, 2\pi),\)
\[\lim_{n \to \infty} d_n(\theta) = d(\theta).\]
True or False? Let \(L(f_n)\) represent the length of the Stage \(n\) curve, and \(L(f)\) represent the length of the unit circle. Then
\[\lim_{n \to \infty} L(f_n) = L(f).\]
True or False? The derivatives of the curves converge uniformly to the derivative of the circle.
Details. Let \(d_n(\theta)\) be the derivative \(\frac{dy}{dx}\) of the Stage \(n\) curve at the point where it intersects a ray making an angle of \(\theta\) with the positive \(x-\)axis, and let \(d(\theta)\) be the derivative \(\frac{dy}{dx}\) of the unit circle at the point where the same ray intersects the unit circle.
True or False? For each \(\theta\) in the interval \((0, 2\pi),\) the sequence \(\{d_n(\theta)\}\) converges uniformly to \(d(\theta).\)
Note. Uniform convergence in this example is equivalent to:
For every \(\epsilon > 0,\) there is a natural number \(N_{\epsilon}\) such that
\[n>N_{\epsilon} \Rightarrow \left|d_n(\theta) - d(\theta) \right| < \epsilon \text{ for all } \theta \text{ in } (0, 2\pi).\]
In which case is the convergence uniform and the lengths of the curves converge to the length of the unit circle?