SCHWEITZER 1953 Problem 1

Let $a_{v}$ and $b_{v}$ , ${v= 1,2,\dots,n}$ , be real numbers such that $a_{1}\geq a_{2} \geq a_{3}\geq\dots\geq a_{n}> 0$ and $b_{1}\geq a_{1}, b_{1}b_{2}\geq a_{1}a_{2},\dots,b_{1}b_{2}\dots b_{n}\geq a_{1}a_{2}\dots a_{n}$ Show that $b_{1}+b_{2}+\dots+b_{n}\geq a_{1}+a_{2}+\dots+a_{n}$ (S. 4)