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Let $n \in \mathbb{N}$, and let $a_1, ... , a_n > 0.$

Show that:

inequality

I got the hint that we have to use this induction step for the induction proof:

Hint

And thats what I got so far in the Induction step but i only come to dead ends:

My first try:

first try first page

first try second page

And my second try:

second try

I hope you can help me or give me a hint and thank you in advance for the answers.

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    $\begingroup$ In the hint, $k$ is doing double duty in the notation. Can you clarify if the inner summation should be $ \sum_{k}^n a_k$ or $ \sum_{i=1}^n a_i$? $\endgroup$
    – Calvin Lin
    Commented Nov 12, 2021 at 1:41

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