Why is $\cos(\cos(\cos(0))) = 0.739$ (with infinite $\cos()$)?

Question

I was calculating the field of view for some telescopes, but then I came to this interesting expression:

$$\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(0))))))))))) = 0.739$$

where we have an infinite number of $\cos()$ and terms are expressed in radians.

I first discovered, that it has a limit, and then I made a program in Python for visualizing this:

import matplotlib.pyplot as plt
import numpy as np
import math
end = 100
fig, ax = plt.subplots()
x = np.arange(0, end)
y = [0]
i = 1
temp = 0
while i < end:
    temp = math.cos(temp)
    y.append(temp)
    i += 1
ax.plot(x, y)

and it's clear that it has a limit.

So I would like to have proof. I tried to start, but it seems so hard, that I've stuck.

Thank you in advance!