
Limits of sequences 
Properties of
convergent sequences 
Sandwich
theorem (result) or squeeze rule 
Least
upper bound (or supremum, abbrev. lub, sup) and greatest lower bound (infimum, abbrev. glb, inf) 
The
definition of the real number e 






Sandwich
theorem (result) or squeeze rule 
Suppose
that {b_{n}}
is a sequence whose terms are bounded above and below (squeezed between)
by sequences {c_{n}}
and {a_{n}}
respectively,
such that 
a_{n}
< b_{n} < c_{n}
for all n
and if a_{n}
®
L
and c_{n}
®
L,
then b_{n}
®
L. 

Least
upper bound (or supremum, abbrev. lub, sup) and greatest lower bound (or
infimum, abbrev. glb, inf) 
Suppose
{a_{n}}
is a bounded increasing sequence of real numbers, then the least upper
bound of the set { a_{n}_{
}: n
Î
N }
is the limit of {a_{n}},
so we write L
= sup a_{n}. 
Proof:
Suppose L
= sup a_{n}
then, for
given e
> 0 there exists an integer n_{0}
such that 
a_{n0}
> L 
e
or L  a_{n0}
< e. 
Since
{a_{n}}
is increasing a_{n}
> a_{n0}
> L 
e
for all n
> n_{0},
and since L
is the upper bound of the sequence then, L
>
a_{n}
for every n,
therefore 

L

a_{n}
 = L 
a_{n}
< L  a_{n0}
< e
for all n
> n_{0}
that is, a_{n}
®
L. 
Similarly,
if {a_{n}}
is a bounded decreasing sequence of real numbers, then the greatest
lower bound of the set
{ a_{n}_{
}: n
Î
N }
is the limit of {a_{n}},
and we write L
= inf a_{n}. 

The
definition of the real number
e 
We
use the monotonic sequence theorem to prove that the sequence defined
by 


is
increasing and its terms remain less than one fixed number as n
tends to infinity, that is, the sequence converge to the
number e. 
Recall
the binomial expansion theorem 

Let
expand a_{n}
using the above theorem 

Then,
to prove a_{n}_{+1}
> a_{n}, that is,
that given is increasing sequence, we can expand a_{n}_{+1}
the same way. 
Observe
that, while passing from n
to n
+ 1, the a_{n}_{+1}
expression gets one new positive term and, at the same time, all the
differences in parentheses raised as every subtrahend decreased (i.e., denominators
increased
to n
+ 1). Therefore, since a_{n}_{+1}
> a_{n} the
sequence is increasing. 
To
show that all terms of the sequence are less than a fixed number M
we will evaluate the quantity a_{n}
on a suitable
way. 
If
we omit second term (consisting 1/n)
in every parenthesis,
they increase, hence 

If
instead of the factors, 3,
4,
. . . , n
in the denominators, starting from the second, we substitute 2,
the denominators
decreased so the fractions increased, thus 

since
applied is the formula for the sum of the finite geometric series whose
ratio is 1/2
and the first term 1. 
As
the right side still depends on n,
to get an expression independent of n,
the one which holds to all terms, drop
the term 1/2^{n}
in the numerator of the right side, what increases the numerator, such
that the final result is 

Therefore,
as all terms of the sequence are less than M = 3 then, the sequence has
a limit that is not greater
than 3. 
Let
write few terms of the sequence to show how slowly it increases, 

So,
we finally write 










Calculus
contents B 



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