Function Convergence Proofs
** Please see the attached file for the complete problem description **
Let f be a function defined on R and, for each natural number n, define the function f_n by....
Decide whether or not you believe the statement is true.
... |f_n(x) - f(x)|<e for any n>N and any x in R. Hence f_n(x) converges to f(x) uniformly on R.
We consider f(x) = x^2. Then for e=1>0, for any n>0, we can set x = n and we have
|f_n(x) - f(x)| = |f(x + 1/n) - f(x)| = |f(n + 1/n) - f(n)| = (n + 1/n)^2 - n^2
= n^2 + 2 + 1/n^2 - n ^2 = 2 + 1/n^2 > 2 = e
Thus f_n(x) does not converge uniformly on R.