1. Consider the sequence a,-_(1 + (-1)") for all n є N. rove that a (b) Prove that (an) diverges using subsequences...
Prove the following languages are not context-free by using the pumping lemma. {b(n) #6(n + 1) | n є N, n-1} where b(n) is binary representation of n with no leading 0 {b(n) #6(n + 1) | n є N, n-1} where b(n) is binary representation of n with no leading 0
4 Consider the sequence () defined by, (a) Using 2, find r2 and r3 and express the results as true rational numbers. (b) Use induction to show that if xi є Q, then xnE Q for all n є N. (c) Prove, using induction, that if 2 x1 3, then 2 xn 3 for all n є N by showing i) 2 < rn < 3 implies that n+13 ii 2 S n 5/2 implies that 2 n+ i) 5/2...
You have to sort a sequence of N elements. The N elements have N/K subsequences of size K each. The subsequences have the following property: All elements of a subsequence are less than those of the preceding one and greater than those of the following subsequence. Example: A sequence of 6 elements with K = 2 is 6; 10; 11; 26; 17). The subsequences here are (7; 6); 110; 11) and (26; 17): Note that all elements of (7; 6)...
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
Please help with #6 'rove: Given a sequence of n2 +1 distinct integers, either there is an increasing subsequence of n+1 terms or a decreasing subsequence of n +1 terms. 'rove: Given a sequence of n2 +1 distinct integers, either there is an increasing subsequence of n+1 terms or a decreasing subsequence of n +1 terms.
Let (xn) be a bounded sequence of real numbers, and put u = lim supn→∞ xn . Let E be the set consisting of the limits of all convergent subsequences of (xn). Show that u ∈ E and that u = sup(E). Formulate and prove a similar result for lim infn→∞ xn . Thank you! 7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
show all work | 2n-1) 2. Consider the sequence |(n+1)! a) is the sequence monotone increasing or monotone decreasing or neither? b) Find upper and lower bounds for the sequence. c) Does the sequence converge or diverge? (Explain) 3. Determine if the series converges or diverges. If it converges, find its sum. => [-1-] c) Ë ?j? – 1-1 j? +1
2. Let (%)-1 be a bounded sequence and let (h) .1 be a sequence that diverges to oo. Prove that (an +bn)ni diverges to oo
1. Consider the sequence (an) with an = Vn2 + n - n, n = 1,2,3,,.... 1.1) Prove that (an) is an increasing sequence. 1.2) Prove that (an) has an upper bound, and therefore has a limit a 1.3) Find a, the limit of an when n + . 1.4) Using Definition 2.2.3 to prove lim an = a. n->00
2. Suppose that A is an rn x n matrix and b є С". Prove that the linear system CSA, b) is consistent if and only if r(A) = r(Ab) 2. Suppose that A is an rn x n matrix and b є С". Prove that the linear system CSA, b) is consistent if and only if r(A) = r(Ab)