Let's consider the sequence of functions $f_n : [0,2] \to \mathbb{R} $ defined by $f_n(x)= (1+x^n)^{1/n}$. I proved that this sequence converges pointwise to the function: $ f(x) = 1$ if $0\le x \le 1$ and $ f(x)=x$ otherwise. My problem is how can I prove that this sequence converges uniformly. (maybe I can prove that $f_n$ is a Cauchy sequence but it look complicated) Please help me )=
Proving that jte sequence $f_n(x)= (1+x^n)^{1/n}$ converges uniformly
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real-analysis
sequences-and-series
uniform-convergence
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1what did you try? did you try proving it straight out of the definition? those functions on [0,2] are bounded, try to use that. Also, did you prove the series is monotone decreasing for a given X? Try to prove the general case (uniform convergence occurs in all such cases of monotone decreasing sequences of functions) – 2012-10-31