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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}(y

_{0}) = {y[itex]\in[/itex]M:d(y,y

_{0})≤R}, for some arbitrary fixed y

_{0}

1) If A=[itex]\phi[/itex], vacuously true.

2)Let A[itex]\subseteq[/itex][itex]\bar{B}[/itex]

_{R}(y

_{0})

Let x

_{1},x

_{2}[itex]\in[/itex]A.

The d(x

_{1},x

_{2})≤d(x

_{1},y

_{0}) + d(x

_{2},y

_{0})≤2R.

Thus, diam(A)≤2R and we see that A is bounded.

Q.E.D.

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