MHB Understanding the Inequality for Solving Limits with Exponential Terms

[Vali]

Hello!

$$\lim_{n\rightarrow \infty }\frac{1}{n}ln(a^{\frac{n}{1}}+a^{\frac{n}{2}}+...+a^{\frac{n}{n}} ), \ a>1$$
I solved the limit by using the following inequality:
$$a^{n}\leq a^{\frac{n}{1}}+a^{\frac{n}{2}}+...+a^{\frac{n}{n}}\leq n\cdot a^{n}$$
After I applied a $ln$ and $1/n$ I got $lna$
My question is about that inequality.Where does this come from ?How can I prove it ?Should I notice something about the exercise to know I've to use this inequality?
Thanks!

 

[Evgeny.Makarov]

Isn't this inequality obvious?

 

[Vali]

$$a^{n}\leq a^{\frac{n}{1}}+a^{\frac{n}{2}}+...+a^{\frac{n}{n}}$$ this I can see is true, it's obvious
$$a^{n}\leq n\cdot a^{n}$$ like the first one, I can see it's true
$$a^{\frac{n}{1}}+a^{\frac{n}{2}}+...+a^{\frac{n}{n}}\leq n\cdot a^{n}$$ this one,I can't see "how it's true", it's not so clear for me why this is true.

 

[Evgeny.Makarov]

There are $n$ terms in the sum, and each does not exceed $a^n$.

 

[Vali]

I understood!
Thank you for your help! :)