Question

This assignment asks you to prove the following Proposition 1 Let {n} and {n} are two sequences of real numbers and L is a number such that (1.a) un → 0, and (1.b) V EN, -L Swn. We illustrate the proposition. To begin, one can check from the definition that 1/n 0. This fact, plus the arithinetic rules of convergence, generate a large family of sequences known to converge to 0. For example, 11n +7 1 11 +7 3n2 - 5n+1 n 3- + In general, a sequence of the form P(n)/Q(n) in which P(n) and Q(n) are polynomial and the degree of P(x) is strictly less than the degree of Q(x) converges to 0. Consider a sequence which is almost polynomial: 1 9n + 6 sin(n) + 8 cos(n) It should be intuitive that the trigonometric functions do not affect the limit. The dominant terms should be the 9n on top and then denominator factor. Rewrite this as bn=9+ sin(n) + 8 cos(n) We expect the limit to be 9. Formally, it is the difference between brand 9 that matters: 6 sin(n) + 8 cos(n) bn -9= - There is an easy, if sloppy, bound for the numerator. 16 sin(n) + 8 cos(n) <6) sin(n)| + 8cos(n) < 14. Obviously this bound is too high; the sin and cos cannot simultaneously have magnitude 1. But it suffices to give an inequality 16. - 915 14 = 14.5 The righthand side is known to converge to 0. Once proven, proposition allows us to conclude that br 9 without further work.

Answer 1

proposition ti Ket of and and a wale are two sequence of real numbens and Liba number such that (a) Wn to and (4.b) unen, Inn-L1s Wn. and L be the limit af of neal rum ben Let funt be a sequence of the the sequenes fuy. land be a sequence Let us choose & an , (since I be its limitpoint) The applying the limiting definition ondanken Wow take wn= Enalan Then cleansy wno and which sat's tes the other condition also Note we can choose any other sequena hand where lim & 0.