Define fn : [0,1] --> r by the
equationfn(x) = xn. show that the
sequence(fn(x)) converges for each x belongs to [0,1],
but thatthe sequence (fn) does not converge uniformly.

The sequence of functions converges to the function

.

Step-by-step explanation:

The limit exists and converges to zero whenever . But, if the sequence is constant and all its terms are equal to , then converges to . Using this result, consider the sequence of functions defined on the interval by . Then, for all we have that . Now, if , then . Therefore, the limit function of the sequence of functions is

.

To show that the convergence is not uniform consider . For any choose  such that . Then

This implies that the convergence is not uniform.

the radius is 2 times the diameter.

step-by-step explanation: