Prove the inequality using simple induction.

Question

I was solving this question:

Let $A_1,A_2,A_3\cdots A_n$ be a sequence of real numbers satisfying $A_{i+j} \leq A_i + A_j ,\forall i, j \in\mathbb{N}$. Prove that $A_1 + \frac{A_2}2 + \frac{A_3}{3} +\cdots + \frac{A_n}n \geq A_n$.

I was able to prove that $A_1 \geq \frac{A_n}n$ for all $n$ being natural using induction. But after that I am unable to use this fact to prove the required statement.

I don't think that from given inequality, any other inequality will actually prove to be useful, so what am I missing?

Answer 1

For $n=1$ it's true.

Let $a_1+\frac{a_2}{2}+...+\frac{a_n}{n}\geq a_n$ for $n\in\{1,2,...,n\}$.

After summing of these $n$ inequalities we obtain: $$na_1+(n-1)\frac{a_2}{2}+...+\frac{a_n}{n}\geq a_1+a_2+...+a_n,$$ which gives $$na_1+a_1+(n-1)\frac{a_2}{2}+a_2+...+\frac{a_n}{n}+a_n\geq$$ $$\geq(a_1+a_n)+(a_2+a_{n-1})+...+(a_n+a_1)\geq na_{n+1},$$ which gives $$(n+1)\left(a_1+\frac{a_2}{2}+...+\frac{a_n}{n}\right)\geq na_{n+1}$$ or $$a_1+\frac{a_2}{2}+...+\frac{a_n}{n}+\frac{a_{n+1}}{n+1}\geq\frac{na_{n+1}}{n+1}+\frac{a_{n+1}}{n+1}$$ or $$a_1+\frac{a_2}{2}+...+\frac{a_n}{n}+\frac{a_{n+1}}{n+1}\geq a_{n+1}$$ and we are done by induction.