- #1

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Given ##f_n=\frac {x} {1+x^2}-\frac {(1+x^2)x} {1+(n+1)^2x^2}## , prove that ##\{f_n\}_{n \in \mathbb N}## converges pointwise and uniformly to a continuous function on the interval ##[0,1]##

The attempt at a solution.

It's easy to prove that this sequence tends to the function ##f(x)=\frac {x} {1+x^2}## pointwise, I just calculated the limit for a fixed ##x## in the interval. Now, I am trying to prove by hand (without using theorems of sequences of functions defined on compact sets or anything of the sort) that the sequence tends to ##f## uniformly. So, to prove this I have to prove that for a given ##ε>0##, there is some ##N \in \mathbb N## such that for all ##n≥N## ##|f-f_n|<ε##.##|f-f_n|=|\frac {(1+x^2)x} {1+(n+1)^2x^2}|=\frac {(1+x^2)x} {1+(n+1)^2x^2}≤\frac {(1+x^2)x} {(n+1)^2x^2}=\frac {(1+x^2)} {(n+1)^2x}##. So, if I could choose ##N## such that for every ##x \in [0,1]##, ##\frac {(1+x^2)} {x}≤ε(n+1)^2##, I would be done, the problem is I don't know which ##N## would work.