REAL ANALYSIS NOTES BY S.K RATHORE
NOTES FOR CSIR NET MATHEMATICS
NOTES FOR IIT JAM MATHEMATICS
We introduce the set of real numbers in an informal way. Once this is done,
the most important property of the real number system, known as the least
upper bound property, is introduced. We assume that you know the set N of
natural numbers, the set Z of integers, and the set Q of rational numbers. You
know that N ⊂ Z ⊂ Q. You also know the arithmetic operations, addition and
multiplication of two natural numbers, integers, and rational numbers. You also
know the order relation m < n between two integers, more generally between two
rational numbers.
Sequences arise naturally when we want to approximate quantities. For in-
stance, when wish to use decimal expansion for the rational number 1/3 we get a
sequence 0.3, 0.33, 0.333, . . .. We also understand that each term is approximately
equal to 1/3 up to certain level of accuracy. What do we mean by this? If we want
the difference between 1/3 and the approximation to be less than, say, 10−3
, we
may take any one of the decimal numbers 0.
3 . . . 3 where n > 3. Or, if we want
to use decimal expansion for √
2, we look at
1.4, 1.41, 1.414, 1.4142, 1.41421, . . . , 1.4142135623730950488016887242 . . . .
The idea of a sequence (xn) and its convergence to x ∈ R is another way of
saying that we give a sequence of approximations xn to x in such a way that if
one prescribes a level of accuracy, we may ask him to take any xn after some
N-th term onward. Each such will be near to x to a desired level of accuracy.
It is our considered opinion that students who master this chapter will begin
to appreciate analysis and the way the proofs are considered.
the most important property of the real number system, known as the least
upper bound property, is introduced. We assume that you know the set N of
natural numbers, the set Z of integers, and the set Q of rational numbers. You
know that N ⊂ Z ⊂ Q. You also know the arithmetic operations, addition and
multiplication of two natural numbers, integers, and rational numbers. You also
know the order relation m < n between two integers, more generally between two
rational numbers.
Sequences arise naturally when we want to approximate quantities. For in-
stance, when wish to use decimal expansion for the rational number 1/3 we get a
sequence 0.3, 0.33, 0.333, . . .. We also understand that each term is approximately
equal to 1/3 up to certain level of accuracy. What do we mean by this? If we want
the difference between 1/3 and the approximation to be less than, say, 10−3
, we
may take any one of the decimal numbers 0.
3 . . . 3 where n > 3. Or, if we want
to use decimal expansion for √
2, we look at
1.4, 1.41, 1.414, 1.4142, 1.41421, . . . , 1.4142135623730950488016887242 . . . .
The idea of a sequence (xn) and its convergence to x ∈ R is another way of
saying that we give a sequence of approximations xn to x in such a way that if
one prescribes a level of accuracy, we may ask him to take any xn after some
N-th term onward. Each such will be near to x to a desired level of accuracy.
It is our considered opinion that students who master this chapter will begin
to appreciate analysis and the way the proofs are considered.
