Let fnx x 1n fx x for n elementof N x elementof R Then fn

Let f_n(x) = x + 1/n, f(x) = x for n elementof N, x elementof R. Then f_n rightarrow f uniformly on R, but it is false that f^2_n rightarrow f^2 uniformly on R. Of course f^2_n rightarrow f^2 pointwise on R.

Solution

Solution :

Given that   f_n(x) =   x + 1/ n and   f(x) = x   for al n E N , x E R .

To show that f_n(x) --> f uniformly on R.

Consider | f_n - f |   = | x + 1/n - x | = | 1 / n|

As n ---> infinity   limit | f_n - f | --> 0 there fore   f_n is converges to f uniformly.

Now we have to show that( f_n)^2 converges pointwise to ( f )^2  

Consider   |(f_n)^2 - f^2 |    = | ( x + 1/n)^2 - x^2 | = | x^2 + 2x / n + (1/ n)^2 - x^2 | = | 2x / n + ( 1/ n)^2 |

this goes to 0 when n ---> infinity eacept at x = 0.

But for x = n   we have | (f _ n)^2 - f^2 | tends to 2 as n tends to infinity.

there fore (f_n)^2 converges to f^2 pointwise but not uniform.