an= 0 in first line 1. Suppose that {Tm} and {a,.) are two sequences such that...
Let (In), and (yn).m-1 be sequences such that Pr – yn| < 1/n for all n. Use the definition of convergence to prove that, if (2n)_1 is convergent, then (Yn)-1 is convergent.
+1 4. Consider the function ISO 0<<1 -1 = 1 0 1<*52 (x - 2)2 => 2 (a) (10) Use the definition of the limit of a function at a point to evaluate with proof (b) (10) Use the definition of continuity at a point to prove that /(x) is not continuous at -1. (e) (2) Is /(x) uniformly continuous on (-1,2)? If it is, prove it. Other- wise, explain why not. (d) (8) Is f() uniformly continuous on (1,3)?...
Un=- V = Exercise 6: Let (Un) and (Vn) be two sequences such that: U. <V. aUn-1 + BVn-1 -1. 0<B<a atß. aVn-1 + BUn-1 atß 1. Let Wn = Un - Vn. Prove that Wn is a geometric sequence. Identify q and V. 2. Prove that (Un) is an increasing sequence and that (Vn) is decreasing. 3. Deduce that (Un) and (Vn) are adjacent sequences. 4. Find the limit l in terms of U, and Vo.
5. Use Rice's Theorem to prove the undecidablity of the following language. P = {< M > M is a TM and 1011 E L(M)}.
for every n. Prove: If (a) converges, then 11. Let (a.) and (b) be sequences such that a, b, < so does (bn). There are several ways to prove this; at least one doesn't involve Cauchy sequences or e. Be careful though you don't know that () converges so make sure that your method of proof doesn't in fact require (b) to converge.
[3] 5. Suppose that f: D[0, 1] → D[0, 1] is holomorphic, prove that \f'(x) < 1/(1 - 1z| for all z e D[0, 1]. [3] 5. Suppose that f: D[0, 1] → D[0, 1] is holomorphic, prove that f'(x) < 1/(1-1-12 for all z e D[0, 1]
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 (...
8. (15) Let REPEATTM = { <M>M is a TM, and for all s € L(M), s = uv where u = v}. Show that REPEATM is undecidable. Do not use Rice's Theorem.
(h) Define f : [0, 2] + R by 122 if 0 <<<1 f(x) = { ifl<152 Using the limit definition of the derivative and the sequence definition of the limit prove that f'(1) does not exist.