Let g_n : R -> R be given by gn (x) := cos2n (x), does gn' converge uniformly?
The derivative is as follows; -2nsin(x)cos2n-1, which I have found converges pointwise to the 0 function.
Formal definition of Uniform Convergence;
For all e>0, there exits N(e) in the natural numbers, such that for all n >= N, for all x in R,
imples Mod( fn(x) - f(x)) < e
where f is the limit of the sequence of functions fn
The Attempt at a Solution
I have been struggling at seeing how to prove functions are uniformly convergent in general, but have been told that this does not converge uniformly, however I really cannot see how this is true, as
Supremum norm(gn' - 0) ->0 as n-> Infinity,
which is an equivalent definition of uniform convergence, because the term cos2n-1 is dominant over 2n.
Thanks for any help :)