Question 182
Let $a_1, a_2, a_3, \dots$ be a sequence of positive numbers. If there exists a positive number $M$ such that for every $n = 1, 2, 3, \dots$,
$$
a_1^2 + a_2^2 + \dots + a_n^2 \lt M a_{n+1}^2,
$$
then prove that there exists a positive number $M'$ such that for every $n = 1, 2, 3, \dots$,
$$
a_1 + a_2 + \dots + a_n \lt M' a_{n+1}.
$$
$$
a_1^2 + a_2^2 + \dots + a_n^2 \lt M a_{n+1}^2,
$$
then prove that there exists a positive number $M'$ such that for every $n = 1, 2, 3, \dots$,
$$
a_1 + a_2 + \dots + a_n \lt M' a_{n+1}.
$$