<s>[INST] Would this solution of the limit of the sequence be correct? Let's suppose that I have the sequence $a_n = \frac{1}{n^2} + \frac{2}{n^2} + \frac{3}{n^2} + \ldots + \frac{n}{n^2}, n \in \mathbb{N}$. And I have to find the limit of the sequence as $n \rightarrow \infty$. Would the below solution be correct?
The sequence $a_n$ can be rewritten as
\begin{align}
a_n &= \sum_{k=1}^{n} \frac{k}{n^2} \\
&= \frac{1}{n^2} \sum_{k=1}^{n} k \\
&= \frac{1}{n^2} \cdot \frac{n(n+1)}{2} \\
&= \frac{n+1}{2n}.
\end{align}
Thus we have
\begin{align}
\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} \frac{n+1}{2n} = \lim_{n \rightarrow \infty} \frac{n}{2n} = \frac{1}{2}.
\end{align}
 [/INST] This seems correct. Personally I would not say $\lim\frac {n+1}{2n}=\lim\frac n{2n}$. Not without explanation at least (e.g. include the step $\lim \frac{n+1}{2n}=\lim \left(\frac{n}{2n}+\frac 1{2n}\right)$).  I would rather say $\lim \frac {n+1}{2n}=\lim \frac{1+\frac1n}{2}$. But this is of course a matter of preference.
</s>