Prove that $f_n(x) = n / (nx + 1) $ is pointwise convergent on $(0,1)$ Prove that $f_n(x) = n / (nx + 1) $ is pointwise convergent on $(0,1)$.


I know that the pointwise limit will be $1/x$ but I want to do it using the $\epsilon$ definition of convergence of a sequence of functions.
I think its pointwise limit is $1/x$ for every $x \in ]0,1[$.
To justify this, taking any $x \in ]0,1[$ and any $\varepsilon &gt; 0$. We have
$$
\big| \frac{n}{nx+1} - \frac{1}{x} \big| = \frac{1}{nx^{2}+x} &lt; \frac{1}{nx^{2}} &lt; \varepsilon
$$
if $n &gt; 1/\varepsilon x^{2}$,
so taking $N := \lceil \frac{1}{\varepsilon x^{2}} \rceil$ suffices.
Given $\epsilon&gt;0$ and $x\in (0,1)$ choose $N\in \mathbb{N}$ as $N = $ (some function of $\epsilon$ and $x$).  Now for a $n&gt;N$ we have 
\begin{align*}
 |f_n(x) - 1/x| = \left| \frac{n}{nx+1} - 1/x \right| = \frac{1}{x(1+nx)}\le \frac{1}{x(1+N x)} \le \cdots \le \epsilon
\end{align*}

So you need to find the function and fill in the ellipses.