Why is it that we can start with a number (say, 1), and press cos x repeatedly, we can get the solution of cos x = x?

e.g. cos (1) = 0.54

cos (0.54) = 0.85

cos (0.85) = 0.65

cos (0.65) = 0.79

cos (0.79) = 0.71

cos (0.71) = 0.76

cos (0.76) = 0.72

.....

until it converges to 0.73