A. Let (X, d) be a metric space. Define a diameter of a subset A of X and then the diameter of an open ball with center at x0 and radios r&gt;0 B. Use (A) to show that every convergence sequence in (X, d) is bounded.

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A. Let (X, d) be a metric space. Define a diameter of a subset A of X and then the diameter of an open ball with center at x0 and radios r&amp;gt;0  B. Use (A) to show that every convergence sequence in (X, d) is bounded.

enlacamig · Accepted Answer

Step 1A) Let&amp;nbsp;(X,&amp;nbsp;d)&amp;nbsp;be a metric space A be a subset of x.We define diameter of A to beSup&amp;nbsp;{d,&amp;nbsp;(x,&amp;nbsp;y)∣x,&amp;nbsp;y∈A}Let&amp;nbsp;∪&amp;nbsp;be an open boll with center at&amp;nbsp;x0&amp;nbsp;and radios&amp;nbsp;r&amp;gt;0Therefore, diameter of&amp;nbsp;∪=2rWe choose x, y to be diametrically opposite two points whose midpoint is&amp;nbsp;x0Next&amp;nbsp;d(x,&amp;nbsp;y)&amp;nbsp;at fains 2r.Hence, for any&amp;nbsp;x,&amp;nbsp;y∈∪d(x,&amp;nbsp;y)≤d(x,&amp;nbsp;x0)+d(x0,&amp;nbsp;y)≤r+r=2r∴&amp;nbsp;diam(∪)=2rStep 2B) Let&amp;nbsp;(xn)n∉N&amp;nbsp;be a sequence which is convergent in&amp;nbsp;(x,&amp;nbsp;d)We assume that, it converges to&amp;nbsp;x∈xChoose&amp;nbsp;∑=1Then&amp;nbsp;∃N∉Ns.t&amp;nbsp;∀h≥&amp;nbsp;d(xn,&amp;nbsp;x)&amp;lt;1Therefore, we haveN1=Sup{d(x,&amp;nbsp;x1),d(x,&amp;nbsp;x2)…,d(x,&amp;nbsp;xn−1),1}&amp;nbsp;Then for any&amp;nbsp;xn&amp;nbsp;in the sepd(xn,&amp;nbsp;x)&amp;lt;1Thus,&amp;nbsp;(xn)n∉N&amp;nbsp;is bounded

Cassandra Ramirez · Accepted Answer

Step 1  Consider the discrete metric d on a set X:  d(x, y)={0 ifx=y1 ifx≠y  Consider the ball of radius r=12 centered at x  Then B(x, r)={x}  Now by definition, diam   A=sup{d(a,b):a,b∈A}  Applying it to our case where  A=B(x,r)  we have diameter of A equal to 0

nick1337 · Accepted Answer

A fairly common example that isn&#039;t completely trivial is the British Rail metric on R, where  d(x, y)=|x|+|y| (or 0 for x=y), so B(x, r) Has diameter 0 for x&amp;lt;r, diameter r for x&amp;lt;r&amp;lt;2x and diameter 2(r-x) for r&amp;gt;2x

A. Let (X,\ d) be a metric space. Define a

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