No because if n is 1 they are equal., more importantly though even if n is not 1 you can still put less than or equal to since it's weaker to be safe, remember less than or equal to does mean less than....or....equal to, so if you use it, it does mean less than but also allows for equality😃
Does capital 'N' have to be a positive integer? I'm using Analysis: with introduction to proof by Lay, and he writes that a sequence converges provided that for each epsilon > 0, there exists a real number 'N' such that for any natural number 'n', n > N implies that | s_n - s | < epsilon.
Good question I can see why it would bother you, I don't like the variation in definitions either lol, math is already hard. It happens a lot too in other areas of math.
@@TheMathSorcerer Thank you! I've seen 'N' defined as a positive integer more often (Rudin, Pugh, Krantz), so I'll probably stick with that. And actually, I looked at a newer edition of Lay's Analysis (I was using the 4th, and I looked at the 5th), and the definition was changed so that 'N' is also defined as a positive integer.
@@TheMathSorcerer Sir in the sequence ( -1) ^n . I set epsilon as 2 and solved. At the end I got limit of sequence value within a range. But when i take epsilon ≤ 1 I don't get any intersection. Why so? Plz reply
Try this vid instead it's better I think ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-b0i-DP-4Xbs.html also there are more in my advanced calculus playlist😄😄
I don't see how your +1 thing is necessary... n needs to be greater than (6/eps)^2. As these are sequences were are dealing with n needs to be greater than the next integer greater or equal to (6/eps)^2. That integer is ceil((6/eps)^2). I don't see where we need the +1 in that.
