Can you give me a few examples of unbounded sequences of functions which are uniformly convergent. I don't quit get what's the main difference between point-wise and uniform convergence: the notion of "converging at the same rate" is not very clear for me.
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Consider the sequence $$ f_n(x) = e^x +\frac1n,$$ that converges uniformly on $\mathbb R$ to the function $f(x)=e^x$. You can replace $e^x$ with any function you want to produce examples of uniformly converging sequences.
Giuseppe Negro
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From Wikipedia: Intuitively, a sequence of functions converges uniformly to f if, given any natural e, we can find an N so that the functions all fall within a "tube" of width (f(x)-e, f(x)+e) for the entire domain of the function. – Tmr Oct 20 '16 at 21:03
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@Tmr: OK, but $\epsilon$ (the number you call $e$) is not a natural, it is a positive real number. – Giuseppe Negro Oct 21 '16 at 09:11
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No, for example $f_n(x) = \frac{1}{n}$ when $x\in [0,1]$ and $f_n(x) = x$ when $x>1$. Then $f_n(x)$ converges uniformly to the function $f(x) = 0$ when $x\in [0,1]$ and $f(x) = x$ when $x>1$.
danman
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