Tuesday, 26 November 2019

real analysis - Showing convergence of the sequence of functions defined by fn=frac1nx+1





Let fn:(0,1)R:x1nx+1



Does (fn)n0 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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