Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.
No, the continuous image of a Cauchy sequence is not necessarily a Cauchy sequence. To understand this, we first recall that a Cauchy sequence (xn)(x_n) in a metric space XX is one where, for every ________ _____ ____ ___ ___ _______.
_____ ________ _____ ______ _____ _____ _________.
___ ______ ___ ___ ____ _____ _____ _________ _____ ___ __________.
________ _____ ___ ____ ____ ____ _________ _______ __________.
______ _______ ______ ________ ____ _______ ___ _________ ___ ____.
_________ __________ __________ ___ ______ _____ _______.
_____ ___ ___ ____ _________ ________ _________ __________ ____ ________.
____ ___ _____ _________ ______ ______ ____ _________ ___ ______ ______ ______.
______ ______ ________ __________ ___ _______ ______ _______.
______ ____ _____ ________ ___ __________ _________ ____ ___ __________ __________ ________.
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___ _______ _______ ______ ______ ___ __________.
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State whether the following statements are true or false. Justify your answers.
a) The outer measure m* of the set
b) A finite subset of a metric space is totally bounded.
c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.
d) The surface given by the equation x+y+z-sin(xyz) = 0 can also be described by an equation of the form z = f(x, y) in a neighbourhood of the point (0,0).
e) A real valued function f on [a,b] is continuous if it is integrable on [a,b].
Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set D U E where
Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.
Verify the hypothesis and conclusions of the Fatou's lemma for the sequence {fn} given by
Give an example of a family f₁ of subsets of a set X which has finite intersection property. Justify your choice of example.
be given by
Show that f is locally invertible at all points in \ {(0,0,0)}.
Which of the following subsets of R are compact w.r.t. the metric given against them. Justify your answer.
i) A = (1,0) in of −
with standard metric.
ii) A = [4,3] − with discrete metric.
iii) {(x, y) ∈ y > 0} −
with standard metric