Verify the inequality!
APL(Dyalog Unicode), 21 bytes SBCS
(+/⊢×1∘⌽)⎕←(⊢÷+/)?⎕⍴0

Try it on APLgolf!
A tradfn submission which prints the list and returns the sum.
                  ⎕⍴0 ⍝ create a vector with input-many 0's
                 ?    ⍝ for each 0, get a random number between 0 and 1
            ⊢÷        ⍝ divide each number by
              +/      ⍝ the sum of all numbers
         ⎕←           ⍝ print the resulting vector
   ⊢×                 ⍝ multiply the vector element-wise
     1∘⌽              ⍝ with the vector rotated to the left by 1
 +/                   ⍝ take the sum of all products

If appending the sum to the list is fine as output, this could be 20 bytes: x,+/x×1⌽x←(⊢÷+/)?⎕⍴0