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Hodgey8806
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Homework Statement
Let M be a metric space and A[itex]\subseteq[/itex]M be any subset:
Prove that if A is contained in some closed ball, then A is bounded.
Homework Equations
Def of closed-ball: [itex]\bar{B}[/itex]R(x) = {y[itex]\in[/itex]M:d(x,y)≤R} for some R>0
Def of bounded: A is bounded if [itex]\exists[/itex]R>0 s.t. d(x,y)≤R [itex]\forall[/itex]x,y[itex]\in[/itex]A
Empty set is defined to be bounded in for these problems
The Attempt at a Solution
Spse that A is contained in some closed ball.
Let that ball be [itex]\bar{B}[/itex]R(y0) = {y[itex]\in[/itex]M:d(y,y0)≤R}, for some arbitrary fixed y0
1) If A=[itex]\phi[/itex], vacuously true.
2)Let A[itex]\subseteq[/itex][itex]\bar{B}[/itex]R(y0)
Let x1,x2[itex]\in[/itex]A.
The d(x1,x2)≤d(x1,y0) + d(x2,y0)≤2R.
Thus, diam(A)≤2R and we see that A is bounded.
Q.E.D.
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