9. Suppose {X,.) are iid, EIX1く00, EX| = 0 and suppose that {cal is a bounded sequence of real numbers. Prove ή 2....
We are given that (an)nen is a sequence of real numbers that is bounded from above. Now consider L:= lim supnoo An. Prove that: For all € > 0 there are infinitely many n for which an > L - € AND For all e > 0 there are only finitely many n for which an > L te
We are given that (an)nen is a sequence of real numbers that is bounded from above. Now consider L:= lim supnoo An. Prove that: For all € > 0 there are infinitely many n for which an > L - € AND For all e > 0 there are only finitely many n for which an > L te
2. Let A be a nonempty set of real numbers bounded above. Define Prove that -A is bounded below, and that inf(-A) = -sup(A). -A={-a: aEA . (5 marks) (You may use results proved in class.) A = 0 , A is bounded above.
1. Let {n} be a sequence of non negative real numbers, and suppose that limnan = 0 and 11 + x2 + ... + In <oo. lim sup - n-00 Prove that the sequence x + x + ... + converges and determine its limit. Hint: Start by trying to determine lim supno Yn. What can you say about lim infn- Yn? 3 ) for all n Expanded Hint: First, show that given any e > 0 we have (...
3) Let (an)2- be a sequence of real numbers such that lim inf lanl 0. Prove that there exists a subsequence (mi)2-1 such that Σ . an, converges に1
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
please do both question clearly,thank
you!
Consider the sequence of real numbers 13. 1 1 2' 1 2 + 2 + 1 2 + Show that this sequence is convergent and find its limit by first showing that the two sequences of alternate terms are monotonic and finding their limits. Prove that any sequence in hence we may suppose that each subsequence has a least term.) (Note that this result and the theorem on the convergence of bounded monotonic sequences...
(15 points) Suppose that a sequence {{n}.00, of real numbers satisfies 52n+1 = 3xn + 2 for all n E N. Show that {{n}", converges. What is limnto Xn? Explain why? the following four nrohlems
Suppose that A is diagonalizable and all eigenvalues of A are
positive real numbers. Prove that det (A) > 0.
(1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
1,2
Let (an)nen be a sequence of real numbers that is bounded from above. Consider L := lim suPn7o An, prove that: For all e > 0 there are only finitely many n for which an > L + €. For all e > 0 there are infinitely many n for which an > L - €.