Define $f_n:[0,1]->\mathbb{R}$ by $f_n(x)=x^n$. Show that the sequence $(f_n(x))$ converges for each $x \in [0,1]$ but the sequence $(f_n)$ does not converge uniformly.
Uniform Convergence versus pointwise Convergence
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calculus
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0Hint for a slick proof: If a sequence of continuous functions converges uniformly what can you say about the limit function? What is the limit function here? Hint for a direct proof: What is the definition of uniform convergence? Draw some pictures and see where things go wrong. – 2012-02-13