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1.3.3_ Proposition. Let {an } be a sequence in R and let a € R -(a) (n 57 a ifand only iffor every € > 0 there is an N such that lan L a| < â‚...

Question

1.3.3_ Proposition. Let {an } be a sequence in R and let a € R -(a) (n 57 a ifand only iffor every € > 0 there is an N such that lan L a| < € for all n 2 N.(b) an6 ifand only if (ag ~a) = 0.

1.3.3_ Proposition. Let {an } be a sequence in R and let a € R - (a) (n 57 a ifand only iffor every € > 0 there is an N such that lan L a| < € for all n 2 N. (b) an 6 ifand only if (ag ~a) = 0.



Answers

If $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ are two sequences, we write $\left\{a_{n}\right\}=\left\{b_{n}\right\}$ if and only if $a_{n}=b_{n}$ for all $n \in N .$ In Problems $59-62,$ use mathematical induction to show that $\left\{a_{n}\right\}=\left\{b_{n}\right\}$. $$ a_{1}=2, a_{n}=3 a_{n-1} ; b_{n}=2 \cdot 3^{n-1} $$

So we had the following series of A. And B. And we're trying to prove that they are the same. So we have a formula for B that is just doing and for a You have a step by step one so to prove that we already have the base case for A. We're trying to grow this, trying to grow that s event is able to to end. So let's just prove the case is okay. That's going to be equal to um Thank you for ya. A. So it's going to be well we just let's assume what does take an argument that S. F. K. Is you go to go ahead And then what is going to be a circe plus 1? Well that's going to be based on a formula is okay. It's kind of like cute. Let me rewrite it eighth. Um A Last two which is impact to N plus two. Uh not too into K-plus two which is what we will get from the formula up here if you plug in case last month. So we have proven the case of K plus one and we have a base case. So based on induction

So we have that a seven is equal to a uh eggs equal to be open two different series. And we need to prove that use an induction. So we have that the event is just to to Republic two and -1. That's the general formula. and four aces different. We have like an initial number And the formula from one number to Phoenix. So to prove that you are the same, Let's use induction. You have a base case already hazel 12 And that does match ever case for B. If we have b. A one you kind of get to to the power of 2 -1 which is super. The base case is good as in the case of a sub K. Right? Let's just say for the sake of argument that is AAA still to the power of two K minus von And let's look at the 8th. Okay. Plus one. So based on what we have is gonna be, what was it 2 to the Power two or 4? Interview for ace of an minus what? Or a self in this case K plus or minus one. Okay, it's going to be four times 2 to the power of two K -1. Which is the in fact going to be two to the power of two K. Last one which is correct. That's what we're going to get. If you plug in K plus one into this formula, you can check it for yourself. So using induction we're done the two series are the same

So we need to prove if we have series A and series B. That they're the same given a Air one is 1, is that one is one. And for any value of A. Seven Is a self be a self and -1 plus two. And given that we need to protect is equal to this. So given that let's go ahead. So um but we need to prove that A seven is go to two in minus one. Okay so let's we have a base case. Now let's look at A. So Kay let's just say for sake of argument that is equal to this, what is a PSA cave plus one? Well that is going to be uh a sub a bus to which is to k minus one plus two or two K plus. Which in fact matches the following formula ico plug in K plus one. So you're the induction, we have proven that uh the series ace and

So we need to prove that 80 N over A. To the hatred Power a third power of five or eight to the power band is equal to 1/8 to the power and minus five use an induction we say well first off the corporate case ankles to one and that's actually does not allow. So what is this modest value for N. Six. Ok. And close to six. Going to be into the 5/8 to the six which is equal to 8 to 5 or 8- five times a. Is equal to one over a. Which is in fact On over 8 to the Pablo 6 -5. So that's the base case. There you have an he goes okay. And given that, so we are assuming that what I'm about to write is correct. We just make an assumption that this is correct. I'm saying assuming this is correct. We need to prove the following that comes after this. So we assume the eight to the five or eight to the K. Is equal to one over a ah five minus. Okay. Actually through the basic K -5 they're trying to figure out what happens if and is he called U. K. Plus one? Well that's going to be eight or 5 over 82 K plus one. She is in fact 8 to 5 Or a two K Time is born over eight. Am I right? I'm allowed to do that Now this is the call to one over AK -5. I'm gonna have long over A to the power of K -5 Time is going over A, Which will give me 1/82 came on a sport. It is going to be the case if you plug in K plus one to the above formula. So we have proven.


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