Verify Bozano–Weierstrass Theorem for the following sets:
i) Set of non-negative integers.
ii) Interval [-1, ¥]
The Bolzano-Weierstrass Theorem states that every bounded sequence has a convergent subsequence. Let's verify this theorem for the two sets you mentioned:
i) Set of non-negative integers:
The set of non-negative integers is a bounded set since it is contained within the closed interval [0, ∞). Any sequence in this set is trivially bounded since all its elements are non-negative integers. However, not all sequences in this set have a convergent subsequence.
Consider the
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Using the principle of mathematical induction, show that
12 + 32 +5x + ...+ (2n - 1)2 =
Show that the function f : R fi R defined byF(X) = 2X + 7 has an inverse by applying the inverse function theorem. Find its inverse also.
If the partition P2 is a refinement of the partition P1 of [a,b] then L(P1,f) < L(P2,f) and U(P2,f) < U(p1,f) Verify this result for the function f(x) = 2cos x defined over the interval and the partitions p1 =
and p2 =
Show that
is conditionally convergent.
Evaluate:
Check whether the intervals [5,9] and [6,12] are equivalent or not.
Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval [1,0] is an open set.
b) Every bounded sequences is not convergent.
c) The function f : [- 2, 2] fi R defined by f (x) = 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 f :[0,2] fi R defined f (x) = x + [x] is not integrable