<span> The ratio compares two quantities (in this case blue and white parts).
A unit rate (or unit ratio) describes how many units of the first type of quantity corresponds to one unit of the second type of quantity. In our case the first type of quantity is blue parts and the second is white parts, so we should find how many blue parts correspond to one white part.
The ratio is: 5/4
We will obtain the unit ratio if we divide each number with 4 in order to get 1 - as a second quantity.
So, (5/4)/(4/4)= (5/4)/1
The unit ratio is: </span>
<span>(5/4)/1.</span>
The points which are collinear are A,N,X.
Given a graph in which there are three lines and seven points.
We are required to find the three points which are collinear.
Collinear points are basically the points that lie on the same straight line or in a single line. Two or more than two points that lie on a line close to or far from each other are said to be collinear.
The points that line on horizontal line which is from east to west and west and east are A,N,X. Because the points are on a single line that's why they are collinear.
Hence the points which are collinear are A,N,X.
Learn more about collinear points at brainly.com/question/2234342
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BY STUDYING THIS LECTURE, YOU MIGHT JUST ANSWER THIS PROBLEM ON YOUR OWN!
Any convergent sequence in any metric space is Cauchy. I think the easiest way to prove that this sequence is Cauchy is to show that it is convergent in C_b[-1,1], ie, to exhibit an element f in C_b[-1,1] and a proof that ||f_n - f|| goes to 0 as n goes to infinity (where || || denotes the supremum norm).
<span>It is clear that the pointwise limit of the f_n's is the function f(x) = (x^2 + 0)^(1/2) = |x|, so if the f_n's converge in the norm of C_b[-1,1] to anything it will have to be that. We just need to show that ||f_n - f|| does indeed go to 0. For this, note that for any x in [-1,1] we have </span>
<span>f_n(x) - f(x) = sqrt(x^2 + 1/n) - sqrt(x^2) </span>
<span>= [sqrt(x^2 + 1/n) - sqrt(x^2)] [sqrt(x^2 + 1/n) + sqrt(x^2)]/[sqrt(x^2 + 1/n) + sqrt(x^2)] </span>
<span>= [(x^2 + 1/n) - x^2]/[sqrt(x^2 + 1/n) + sqrt(x^2)] </span>
<span>= (1/n)/[sqrt(x^2 + 1/n) + sqrt(x^2)] </span>
<span>and since x^2 >= 0 for all x we have sqrt(x^2 + 1/n) + sqrt(x^2) >= sqrt(0 + 1/n) + sqrt(0) = sqrt(1/n) and therefore </span>
<span>|f_n(x) - f(x)| <= (1/n)/[sqrt(1/n)] = 1/sqrt(n) for all x in [-1,1], </span>
<span>and hence ||f_n - f|| <= 1/sqrt(n) holds for all n. [If you think about this argument for a moment you will see that it actually can be used to show that ||f_n - f|| = 1/sqrt(n) for all n, but this isn't important.] Since 1/sqrt(n) goes to 0 as n goes to infinity it follows that f_n converges to f in the norm of C_b[-1,1] and hence that the sequence f_n is Cauchy in C_b[-1,1]. </span>
<span>You may have noticed there is nothing special about [-1,1] here; for any closed interval [p,q] the same estimate could be used to show that f_n converges to f in the norm of C_b[p,q]. In fact the sequence f_n converges to f uniformly on all of R. [It is not easy to fit this into a metric space context because the most obvious choice of a metric space for discussing uniform convergence on R is the space C_b(R) of bounded functions on R, and while f_n - f is bounded, and hence in C_b(R), for any n, neither of the individual functions f or f_n is bounded, and hence neither f nor any of the f_n's is in C_b(R).] </span>
<span>In general, when asked to prove that a sequence is Cauchy, if you can identify its limit, as we could here, the easiest thing to do is to prove that it's convergent (which implies that it's Cauchy). Proving that a sequence is Cauchy straight from the definitions is more commonly done only when the limit of the sequence cannot be explicitly identified. This isn't as rare as it sounds, e.g. when you are proving general results about convergence you generally do not have 'formulas' for the limit and must proceed in this way. For example, the usual proof of the fact that an absolutely convergent series of real numbers must be convergent does this. One uses the hypothesis that sum |a_n| is convergent to show from the definition of "Cauchy sequence" that the sequence a_1, a_1 + a_2, ..., a_1 + a_2 + ... + a_k, ... is Cauchy, and therefore convergent, and therefore that sum a_n converges. This is done because there is no explicit formula for the sum of an arbitrarily given absolutely convergent series.</span>
