## Transcribed Text

6.
(10
points)
A
function f D R is said to be uniformly approximated
by polynomials if for every e > 0 we can find a polynomial p(x) such that
If(x) p(x) < € for all x € D.
Show that the following two functions cannot be uniformly approximated
by polynomials:
(a) The function f (0,1) -> R given by f(x) = :
(b) The function g R -> R given by g(x) = 1+x²
Hint: For part (b) you may use that a polynomial on R is bounded if and
only if it is constant.
7. (15 points)
(a) Define what it means for a sequence {fn D -> R} to converge uni-
formly to the function f D -> R.
(b) For each n € N, let fn R R be given by fn(x) = Vx2+#
Prove that the sequence {fn} converges uniformly on R to the function
f R -> R given by f(x) = |x|.
(c) Give an example, with proof, of a domain D and a sequence of func-
tions which converges pointwise, but not uniformly on D.
8. (10 points) For each n € N let fn R - R be given by fn(x) = En-on
Let f R + R be given by f(x) = e2. Let r be any positive real number.
Prove that {fn} converges uniformly to f on [-r,r].
Hint: Use the Lagrange Remainder Theorem. You may also use that
limn-+0 rth n! = 0 (Lemma 8.13).

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