Does $$f_n(x) = \frac{x}{1+nx}, \qquad x \in [0,1]$$ converge uniformly on $[0,1]$?
I have shown that $f_n(x)$ converges pointwise to $f(x) \equiv 0$, but I am struggling to prove or disprove uniform convergence.
sequences-and-seriesuniform-convergence
Does $$f_n(x) = \frac{x}{1+nx}, \qquad x \in [0,1]$$ converge uniformly on $[0,1]$?
I have shown that $f_n(x)$ converges pointwise to $f(x) \equiv 0$, but I am struggling to prove or disprove uniform convergence.
Best Answer
Hint
Note that
$$\frac{x}{1+nx}\le \frac{1}{n}, \forall x\in[0,1],\forall n\in \mathbb{N.}$$