Hello!!

I want to check which of the following sequence converges and and if it doesn't converge I want to check if it has a convergent subsequence.
$\displaystyle{d_n=\frac{n^2+1}{n+1}, \ n\in \mathbb{N}}$

We have that\begin{equation*}\lim_{n\rightarrow \infty}d_n=\lim_{n\rightarrow \infty}\frac{n^2+1}{n+1}=\lim_{n\rightarrow \infty}\frac{n\left (n+\frac{1}{n}\right )}{n\left (1+\frac{1}{n}\right )}=\lim_{n\rightarrow \infty}\frac{n+\frac{1}{n}}{1+\frac{1}{n}}=\frac{\infty+0}{1+0}=\infty\notin \mathbb{R}\end{equation*}

@Stefan Perko

Sorry Stefan, I misread the comment. I thought the $\forall$ quantifier was applying to only the first variable "$(\forall x R(x) )\rightarrow B(x)$". I have since changed the suggested edit I made to your version.

Note however that the two expressions can be different:

Consider a finite domain of discourse containing only 3 ravens, in this case:
$\forall x : R(x)\rightarrow B(x)$ may be re-written:
$(R\rightarrow B)_1 \land (R\rightarrow B)_2 \land (R\rightarrow B)_3$

While $R\rightarrow B$ can be re-written: $R_1 \land R_2 \land R_3 \rightarrow B_1 \land B_2 \land B_3$