Cauchy sequence - https://note.com/noppoman/n/ne938ba48e1d5

cauchy-sequence.py

from matplotlib import pyplot as plt
import sympy as sym
from IPython.display import display, Math

x, n, m = sym.symbols('x, n, m') 
expn = sym.sin(x)**2

f = lambda n: sym.expand_trig(expn)/n
F = lambda f: sym.Integral(f, (x, 0, 2))

display(Math("f_n(x) = "+ sym.latex(f(n))))
display(Math("\|f_m - f_n\| = "+ sym.latex(F(f(n) - f(m)))))

epsilons = []
distances = []

n_range = np.arange(1, 301, 2)
m_range = np.arange(2, 301, 2)

for n, m in zip(n_range, m_range):
    epsilon = np.sqrt(1.189/n**2 + 1.189/m**2)
    d = F(f(n) - f(m)).evalf()
    
    epsilons.append(epsilon)
    distances.append(d)
    
plt.figure(figsize=(12, 6))
plt.plot(n_range, distances)
plt.plot(n_range, epsilons, alpha=0.7)
plt.xlabel('n')
plt.ylabel('value')
plt.legend(['‖f_m - f_n‖', 'epsilon'], prop={'size': 15})
plt.show()

plt.figure(figsize=(12, 6))
plt.plot(n_range[100:], distances[100:])
plt.xlabel('n')
plt.ylabel('value')
plt.plot(n_range[100:], epsilons[100:], alpha=0.7)
plt.title("n, m > 200")
plt.legend(['‖f_m - f_n‖', 'epsilon'], prop={'size': 15})
plt.show()