Show your work and give clear explanations and proofs in all problems. If you use a theorem, state the theorem
Show your work and give clear explanations and proofs in all problems. If you use a...
Show your work and give clear explanations and proofs in all problems. If you use a theorem, state the theorem Use the e. 8 definition of the limit of a function to prove that lim #2 +2 = V3.
Instructions: Give clear and detailed proofs in all problems. (provided you state what you are using and verify the hypothesis), unless you are asked to prove a result using the definition. 10.) Prove using the definition of limit that if (an) and (bn) are sequences in R such that (an) converges to zero and (bn) is bounded, then anbn) converges to zero.
PLEASE HELP WITH PROOF!! 8. Let an > 0 for all n in 1. Show that if an converges, then Ĉ vanın converges. [Hint: Expand [van - (1/n)]2.) N =
Please prove this, thanks! 2. Let {xn n21 be a sequence in R such that all n > 0. If ( lim supra) . (lim supー) = 1 Tn (here we already assume both factors are finite), prove that converges.
Prove that is an integer for all n > 0.
Assume that the sequence defined by a1 = 3 an+1 = 15-2·an is decreasing and an > O for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by fn(x):= e-nx. Does {{n}n>1 converge uniformly on [0, 1]? Does it converge uniformly on (a, 1) with 0 <a<1? Does it converge uniformly on (0, 1)?
Use the Principle of Mathematical Induction to prove that (2i+3) = n(n + 4) for all n > 1.
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
3. Use the mean value theorem to prove the following inequality. (1 +x)" >1 for z >0 andnEN