Solution-verification of $\lim_{n\to\infty}\left(\frac{1}{\sqrt[5]{n^5+1}}+\frac{1}{\sqrt[5]{n^5+2}}+\cdots+\frac{1}{\sqrt[5]{n^5+n}}\right)$

Question

I solved this limit and got the solution $1$. I tryed to check the solution on WolframAlpha, but it couldn't generate the solution.

So I wanted check the solution here. This is an exam problem from analysis course I attend.

$$\lim_{n\to\infty}\left(\frac{1}{\sqrt[5]{n^5+1}}+\frac{1}{\sqrt[5]{n^5+2}}+\cdots+\frac{1}{\sqrt[5]{n^5+n}}\right)$$

$$\text{It follows}$$

$$\underbrace{\left(\frac{n}{\sqrt[5]{n^5+n}}\right)}_{\stackrel{n\to\infty}{\longrightarrow}\ 1}\leq\sum_{k=1}^{n}\frac{1}{\sqrt[5]{n^5+k}}\leq\underbrace{\left(\frac{n}{\sqrt[5]{n^5+1}}\right)}_{\stackrel{n\to\infty}{\longrightarrow}\ 1}$$

$$\text{And therefore,}$$

$$\lim_{n\to\infty}\left(\frac{1}{\sqrt[5]{n^5+1}}+\frac{1}{\sqrt[5]{n^5+2}}+\cdots+\frac{1}{\sqrt[5]{n^5+n}}\right)=1$$

My question, is the result correct ?

Thanks in advance

P.S. Should I delete these kinds of questions if the answer to them is a simple yes, since they don't provide much information and may not be of great use to anyone except me ?