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convergence of sequence and inner product as the distance function

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1) We are required to show
liminf limsup
nn
nn
ss
 
Without loss of generality,
n
s
is a bounded sequence. Otherwise, there is no question of
existence of the limit.
So, there exists two real numbers h and k such that
n
h s k
for every
nN
By completeness axiom, we follow that
inf
n
s
and
sup
n
s
Now, we can write that
n
s


for every
nN
Since
are real numbers, by the properties of real numbers, we can easily follow that
n
s

for every
In other words,
=
inf
n
s
and
sup
n
s
Replace
with
liminf
n
n
s

in this equation, we get
liminf limsup
nn
nn
ss
 
Similarly arguing with the real number
, we get
limsup liminf
nn
nn
ss
 
2)
(a) The indicated order of the set is
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1, ,1,1, , , , , ,1,1, , , , ,...,1,1, , ,..., , ,..., ,1,1,...
2 2 3 4 3 2 2 3 4 5 2 3 1 2nn
Observe that the member 1 is repeating after every sequence of palindrome sequence
1
n



So, the sub sequences can be written as
1 1 1 1 1
1 , 1, , ,..., , ,...,
2 3 1 2nn



So, the limits of these sub sequences are
1
lim1 1,lim 0
nn
n
 

So, the limit supremum of
1
is 1 and infimum is also 1.
On the other hand, the limit supremum of
1 1 1 1 1
1, , ,..., , ,...,
2 3 1 2nn



and limit infimum of
1 1 1 1 1
1, , ,..., , ,...,
2 3 1 2nn



is 0.
Putting this information together, the sub sequential limits are 0 and 1.
(b) From the above discussion, the limit supremum of
n
s
is 1 and limit infimum is 0.
3) First, we shall try to understand the concept and then go for the theoretical generalization.
Take the examples
1 1 1 1 1 1
1, , ,... ,..., 1, , ,..., ,...
2 3 2 3
nn
st
nn
 
sup 1,inf 0,sup 0,inf 1
n n n n
s s t s
From these, we can say
   
sup sup 1 0 1,inf inf 0 1 1
n n n n
s t s t
Now,
0,0,0,... 0
nn
st
sup{ } 0,inf{ } 0
n n n n
s t s t
Putting the above results together, we can write that
   
inf inf inf{ } sup{ } sup sup
n n n n n n n n
s t s t s t s t
Now, we shall generalize this result.
Let us take up the necessary condition that we are discussing about the sequences of non
negative terms first.
Also,
 
,
nn
st
are bounded sequences.

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