Final Answer:
A) The sequence of functions
is decreasing and converges uniformly to a limit function f. However,
, illustrating that the Monotone Convergence Theorem does not hold for decreasing functions.
Step-by-step explanation:
In this example, each function
is defined as the characteristic function of the set
, denoted by
. Firstly, we establish that
is decreasing. As n increases, the set
becomes larger, causing
to take the value 1 on a smaller set, hence decreasing.
Secondly, to show uniform convergence, for any
, we can choose N such that for all
,
, where f is the pointwise limit of
. This implies that
converges uniformly to f.
Finally, the failure of the Monotone Convergence Theorem is evident when evaluating the integrals. The integral of the limit function f is not equal to the limit of the integrals of
. This discrepancy arises due to the unbounded nature of the sets
, leading to a failure in the conditions required for the Monotone Convergence Theorem.