As you can see from previous discussion:

1, No matter how many steps you go through, you will never

    repeat any number you have used. So you also avoid going

    through infinite loop;

2. Throughout the whole positive integer axis, the return rate or

     speed is much bigger or faster, and this rate or speed is

     evenly distributed everywhere as well.

Based on these, you have:

1. When k = 1, the condition satisfied;

2. Suppose k = n satisfy the requirement;

3. When k = n + 1, you can try up to infinite number of times,

    the Result Number will reach less than n + 1 at least once.

This proved that

3X + 1 will always return to the minimum is true.

Step 3 takes some times to make sense, nevertheless

it is truly valid.