Let f_(k):[0,1]->R be a sequence of functions such that
(1) |f_(k)(x)|<=M_(1) for all kinN and xin[0,1],
(2) |f_(k)^(')(x)|<=M_(2) for all kinN and xin[0,1].
for some positive M_(1),M_(2).
(a) Prove that there exists a subsequence of {f_(k)} which converges uniformly on
[0,1.]
(b) If the assumption (1) is omitted, can {f_(k)} still have a convergent subsequence?
If yes, prove it; If not, give an counterexample.
(c) Show that the assumption (1) can be replaced by f_(k)(0)=0 for all kinN.
Let f : [0,1] -> R be a sequence of functions such that
(1)|fz(x)|< Mi for all k e N and x E[0,1]
(2)f{(c)|< M2 for all k e N and x E[0,1]
for some positive Mi, M2
(a) Prove that there exists a subsequence of{fk}-i which converges uniformly on [0,1].
(b) If the assumption (1) is omitted, can {fi}-, still have a convergent subsequence?
If yes, prove it; If not, give an counterexample.
(c) Show that the assumption (1) can be replaced by fk(0) = 0 for all k e N.
