角谷猜想的证明
Proof of Angle of valley guess

n0 is an integer number.
Any integer number can be expressed as binary system

If f(n)= n+1 n is an odd number. It still works, then all odd numbers is included. 3n+1 is included.
Equals to f(n) = n -1 the same, because all odd numbers is included.

if n0 is an even number.
If

n0 is an odd number.

n1 = (n_0-1)/2 = a_1*2^0+a_2*2^1+a_3*2^2+...+a_(b-1)*2^(b-1)

So it is the same , else if a_1 =0,
Then n1 is an even number,
n1/2 = a_2*2^0+a_3*2^1+...+a_(b-1)*2^(b-2)
Finally , if a_2,a_3,...,a_(b-2)=0 and a_(b-1) = 1.
n1/2^(b-2) = a_(b-1) = 1.
The final number is 1.

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