1. The problem statement, all variables and given/known data
Show that the sequence of functions ##x,x^2, ... ## converges uniformly on ##[0,a]## for any ##a\in(0,1)##, but not on ##[0,1]##.
2. The attempt at a solution
Is this correct? Should I add more detail? Thanks for your help!
Let ##\{f_n\} = \{x^n\}##, and suppose ##f^n \to f##. We must show that for ##\epsilon>0##, there exists an ##N## such that ##d(f,f^n)<\epsilon## whenever ##n>N## for all ##x.##
For ##a\in (0,1)##, it is clear to see that ##x^n\to 0## as ##n## approaches infinity. We must then show ##|x^n|<\epsilon## whenever ##n## is greater than some ##N##.
On ##[0,a]##, ##x^n## attains its max at ##x=a## so ##x^n<a^n##. Then note that ##a^n## decreases with increasing ##n##, so we choose ##N## such that ##a^N<\epsilon##.
##\{f_n\}## does not converge uniformly on ##[0,1]## because at ##x=1##, ##f^n = (1)^n =1\ne 0## for all ##n##.
Show that the sequence of functions ##x,x^2, ... ## converges uniformly on ##[0,a]## for any ##a\in(0,1)##, but not on ##[0,1]##.
2. The attempt at a solution
Is this correct? Should I add more detail? Thanks for your help!
Let ##\{f_n\} = \{x^n\}##, and suppose ##f^n \to f##. We must show that for ##\epsilon>0##, there exists an ##N## such that ##d(f,f^n)<\epsilon## whenever ##n>N## for all ##x.##
For ##a\in (0,1)##, it is clear to see that ##x^n\to 0## as ##n## approaches infinity. We must then show ##|x^n|<\epsilon## whenever ##n## is greater than some ##N##.
On ##[0,a]##, ##x^n## attains its max at ##x=a## so ##x^n<a^n##. Then note that ##a^n## decreases with increasing ##n##, so we choose ##N## such that ##a^N<\epsilon##.
##\{f_n\}## does not converge uniformly on ##[0,1]## because at ##x=1##, ##f^n = (1)^n =1\ne 0## for all ##n##.
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