12.5.11 Olympic Algebra

Condition: let for the sequence \ (\ left \ {x_ {n} \ right \} \) \ [\ begin {array} {l} x_ {n} = \ sqrt {a_ {1}+\ sqrt {a_ {2}+\ cdots+\ sqrt {a_ {n}}}, \ quad a_ {i}> 1, i = 1.2, \ ldots \\ \ text {and} \ lim _ {n \ rightarrow \ infty} \ left (\ frac {1} {n} \ ln \ left (\ ln a_ {n} \ right) \ ru) <\ ln 2. \ end {array} \] prove that \ (x_ {n} \) converges.