The problem has two parts:
PART A:
Here we must show that the sequence of functions converges to a function f
Since for each xεR and for each ε>0
and by choosing ,then
for all ,n:
we can conlude that the limit is the zero function.
PART B:
Here we must show that the sequence converges uniformly to the zero function:
This can be done by many ways.One of them is by appling the definition of uniform continuity i.e:
given an ε>0 we must find a natural No κ such that:
for all ,n,x: and .
The proof of that is based on the following inequality:
for each real,x and natural,n.
Then we can easily deduce from that inequality that:
,And since
if we choose ,then:
for all ,n,x: and .
Thus the sequence converges uniformly to the zero function.
Another way is to use the bounded sequence of functions theorem .
And since the sequence of functions is bounded ,since for all ,n and,x
because .and also the zero function is also bounded then according to the theorem the sequence of functions converges uniformly to the zero function