Let (Xd) be a metric space and suppose that a sequence (a.} has a subsequence that converges to an element LEX. One of these statements is 5. correct: 1. The sequence {an} itself must converge. 2. The sequence (a.} may not converge. Use your intuition to think of which is true. Then, prove it.

Q: (a) Prove that if the sequence {a„} converges to a, then the sequence {|a„]} converges to |aļ.…

A:

Q: Determine whether the sequence diverges. If it converges, find the Converges cr An- cos ( Pie, aiso…

A: To determine whether the below sequence converges or diverges. an=cosnπn+1

Q: Let {ai} be a sequence in a metric space X,d. Prove that lim an = p if and only if lim d(an, p) =…

A:

Q: Determine whether the sequence converges or diverges. If it converges, find the limit. (If the…

A: A sequence is convergent of it approaches to some limit therefore, find limn→∞n2e-5n it is as…

Q: 2. Let X be a non-empty set. Define the discrete metric on X and answer each of the following: (i)…

A: First we have to define the discrete metric on X. Definition (Discrete Metric): The discrete metric…

Q: Define a sequence of ordered pairs (x;, y;) as follows: (xo, Yo) = (3,1) %3D Xn+1 = 3xn + 4yn, n > 0…

A:

Q: Exercise 10.2.4. In view of Equation (10.2.2), one might suppose that the following is true: (Σ²).…

A: Given equality: ∑i=0nxi·∑i=0nyi=∑i=0nxiyi   a) Verify the equality is always true. If not, give an…

Q: Determine whether the seauence converges or diverges.If it conveges, find the loon 100 l00 + 100 キュ

A: