Y= N/D D≠0
Hence if
denominator is zero then real number can never be possible.
So 1/0, 2/0, 3/0,………..are practically
impossible.
i.e. 1/0=
undefined.
Showing posts with label Real Analysis. Show all posts
Showing posts with label Real Analysis. Show all posts
Archimedean Property of real numbers
Theorem:- Let a be any real number and b any +ve real number. Then there exists a
positive integer n such that
nb>a
Proof:- Given that aϵ R & bϵ R+
So there are two possible cases.
Case 1:-
When a≤0
In
this case the relation nb>a is always true because the value of nb is always
+ve.
Case 2:- When a>0
Let
us that there exists no +ve integar such that nb>a
Then
we have nb≤a ⩝ n∈N
It means
that a be the upper bound of set S which is given by
S= {b,2b,3b,………..} = {nb:n∈N}.
Boundness of subsets of R
Upper bound of a subset
of R:-
Let S be a subset of real numbers. If there exist a real
number K , such that x≤K ⩝ x∈S
Then K is called an upper bound of set S.
If there
exists an upper bound for a set S then it is called “bounded above”.
Example:-
The set S= {………-4,-3,-2,-1} is bounded above & 9 is an upper bound.
Archimedean Property of Real Number
Let be any real number. Then there exists a natural number such that .
This theorem is known as the Archimedean property of real numbers. It is also sometimes called the axiom of Archimedes, although this name is doubly deceptive: it is neither an axiom (it is rather a consequence of the least upper bound property) nor attributed to Archimedes (in fact, Archimedes credits it to Eudoxus).
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