Does $f_n(x) = \frac{x}{1+nx}$, with $x \in [0,1]$, converge uniformly on $[0,1]$?

Question

Does $$f_n(x) = \frac{x}{1+nx}, \qquad x \in [0,1]$$ converge uniformly on $[0,1]$?

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I have shown that $f_n(x)$ converges pointwise to $f(x) \equiv 0$, but I am struggling to prove or disprove uniform convergence.

Answer 1

Hint

Note that

$$\frac{x}{1+nx}\le \frac{1}{n}, \forall x\in[0,1],\forall n\in \mathbb{N.}$$