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

Question

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 1

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

Answer 2

Your proof is correct, and you can use wolfram here to see a nice illustration of this fact.

Answer 3

Yes, your arguments are correct. For your question (2): we have

$|f_n(1/n)|=1/2$. This shows that the convergenc is not uniform, otherwise we would have $|f_n(1/n)| \to 0$ as $n \to \infty$