Delete sublists by a numerical criterion

Question

I am trying to use DeleteCase function but I don't know how to implement it in the way I need.

I have a list of sublists of numbers, for example:

{{1,1,1,1},{1,1,1,2},{4,2,5,1},{5,3,2,4}}

and I want to do is to delete all the sublists that contains an element greater than $4$.

The result would be like this:

{{1,1,1,1},{1,1,1,2}}

I am also avoiding loops or recursion.

Answer 1

 list = {{1, 1, 1, 1}, {1, 1, 1, 2}, {4, 2, 5, 1}, {5, 3, 2, 4}};    
 Select[list, Max[##] < 4 &]

Answer 2

lst = {{1, 1, 1, 1}, {1, 1, 1, 2}, {4, 2, 5, 1}, {5, 3, 2, 4}}; 

Pick[#, UnitStep[4 - Max /@ # ], 1] & @ lst
Pick[#, LessEqualThan[4] /@ Max /@ #]& @ lst 
Cases[lst, _?(Max @ # <= 4 &)]
DeleteCases[lst, _?(Max @ # > 4 &)]
Select[lst, Composition[LessEqualThan[4], Max]]

all give

{{1, 1, 1, 1}, {1, 1, 1, 2}}