(b) Suppose that en is a sequence such that 0 <In < 2011 for all n e N. Does lim an exist? If it exists, prove it. If not, give a counterexample. (c) Suppose that in is a sequence such that 0 < < 21 for all n E N.Does lim exist? If it exists, prove it. If not, give a counterexample. 20
3. (a) Given n e N, prove that sup{.22 : 0<x<1} = 1 and inf{.22n: 0<x<1} = 0. (b) Find the supremum of the set S = {Sn: ,ne N}. Give a proof.
Exercise 17: Let (an) be a sequence. a) Assume an> 0 for all n E N and lim nan =1+0. Show that an diverges. n=1 b) Assume an> 0 for all N EN and lim n'an=1+0. Show that an converges. nal
Suppose a is a real number and 1 + a > 0. Prove that (1 + a)" > 1+ na for every integer n > 1.
= (a) Suppose that limx+c f(x) L > 0. Prove that there exists a 8 >0 such that if 0 < \x – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on [a, b] and f(x) > 0 for all x € [a,b], then there exists an e > 0 such that f(x) > e for all x E [a, b].
Prove that for all integers n > 0, 2 (na + n).
4. (a) Suppose that limz-c f(x) = L > 0. Prove that there exists a 8 >0 such that if 0 < 12 – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on (a, b) and f(x) > 0 for all x € (a, b), then there exists an e > 0 such that f(x) > € for all x € [a, b].
all three questions please. thank you
Prove that for all n N, O <In < 1. Prove by induction that for all n EN, ER EQ. Prove that in} is convergent and find its limit l. The goal of this exercise is to prove that [0, 1] nQ is not closed. Let In} be a recursive sequence defined by In+1 = -) for n > 1, and x = 1. Prove that for all ne N, 0 <In < 1....
(a) Suppose that lim x→c f(x) = L > 0. Prove that there
exists a
δ > 0 such that if 0 < |x − c| < δ, then f(x) >
0.
(b) Use Part (a) and the Heine-Borel Theorem to prove that if
is
continuous on [a, b] and f(x) > 0 for all x ∈ [a, b], then
there
exists an " > 0 such that f(x) ≥ " for all x ∈ [a, b].
= (a) Suppose...
Problem 8 You can assume that L = {a"be": n > 0} is not context free. Prove the following: Show that L-ab: n20 is not context free Show that L = {w E {a,b,c,d)* : na(w) = nb(w)-ne(w) = nd(w)) is not context free Note that na(w) means the number of a's in w .