real analysis - Showing convergence of the sequence of functions defined by $f_n = frac{1}{nx +1}$

Let $f_n: (0,1) \to \mathbb{R}: x \mapsto \frac{1}{nx+1}$

Does $(f_n)_{n\geq0}$ converge pointwise? Uniformly?

My attempt:

Let $x \in (0,1)$. Then $\lim_{n \to \infty} f_n = 0$. Hence, $\forall x \in (0,1): f_n(x) \to 0$ and we deduce that $(f_n)$ converges pointswise to $0: (0,1) \to \mathbb{R}: x \mapsto 0$

Now, since

$$\sup_{x \in (0,1)}\left \vert \frac{1}{nx+1} - 0\right \vert = 1 \to 1 \neq 0$$

it follows that $(f_n)$ does not converge uniformly.

Questions:

(1) Is this correct?

(2) Are there alternatives to show that it is not uniform convergent?

Answer

Yes, it is correct. I think that that way of proving that the convergence is not uniform is the simplest one. However, I would have added an explanation for the equality

$$\sup_{x\in(0,1)}\left|\frac1{nx+1}\right|=1.$$