Verify Bozano-Weierstrass Theorem for the following sets:
i) Set of non-negative integers.
ii) Interval
d) Check whether the limit exists or not?
i) Consider the set of non-negative integers ( \mathbb{Z}_{\ge 0} = {0, 1, 2, 3, \dots } ). The Bolzano-Weierstrass Theorem states that every bounded sequence in ( \mathbb{R} ) has a convergent subsequence. Take any sequence ( a_n = n ) in this set. Clearly, ( a_n ____ _________ ______ _______ ____ _________ ________ _____ ________ __________.
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Find a and b such that exists.
Write the inequality in the modulus form.
Show that , the lagrange's form of remainder in the Maclaurin series expansion of e4x, tends to zero as
. Hence obtain the Maclaurin's infinite expansion for e4x.
Find the following limit
Check whether the set of integers is countable or not.
Show that is an algebraic number.
Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval ]0, 1] is an open set.
b) Every bounded sequences is not convergent.
c) The function defined by
is uniformly continuous.
d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.
e) The function defined by
is not integrable.