Let fn:(0,1)→R:x↦1nx+1
Does (fn)n≥0 converge pointwise? Uniformly?
My attempt:
Let x∈(0,1). Then lim. 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.
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