Is Uniform Convergence Implying Boundedness of the Limit Function?

dracond
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Homework Statement



Just trying to get feel for uniform convergence and it's relationship to boundedness. If a sequence of functions (fn) converges uniformly to f and (fn) is a sequence of bounded functions, is f also bounded?

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The Attempt at a Solution

 
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If the convergence is uniform then for all e there is an N such that |fn(x)-f(x)|<e for all n>=N. If fN is bounded and f unbounded, how can this be?
 
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Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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