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Hello!

I've got a short question to an example.

I should check the following sequence for uniform convergence on the whole of [itex]\mathbb{R}[/itex]:

[itex]f_n(x)=\frac{nx(7+sin(nx))}{4+n^2x^2}[/itex]

It says that the conevergence is nonuniform, because:

[itex]sup_{\mathbb{R}}|\frac{nx(7+sin(nx))}{4+n^2x^2}|\ge{\frac{7+sin1}{5}[/itex]

Obviously they put [itex]x=1/n[/itex]. I cannot see why this is true.

I tried to differentiate the given function sequence and see, wether the derivative can become zero or not, i.e. I look for maximum values, but the derivative term gets difficult to manage in the end.

How can I make it clear to myself that the above inequality holds?

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# Homework Help: Uniform conevergence

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