Let I be the interval 0 infinity For n elementof N and t ele

Let I be the interval [0, infinity). For n elementof N and t elementof I, let f_n(t) = sin(squareroot t + 4n^2 pi^2). Show that the sequence {f_n} is equicontinuous on I; Show that {f_n} does not contain a subsequence which is uniformly convergent on I.

Solution

1) we want to show that {f_n} is equicontinuous on I that is we want each term to be continuous and on every neighbourhood its variation is same now as sin is periodic function and in fact is continuous on whole of R so each term is continuous follows from fact that sin is periodic function and in fact is continuous on whole of R

so sequence {f_n} is equicontinuous on R and in particular on I

2)suppose on contrary it contains some subsequence say {f_n0} of{f_n} which is uniformly continuous but then each would be uniformly continuous which contradict to fact that as n becoming large f_n is not boundd so we are done .