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Prove that the closure of set is bounded if the set itself is bounded.

I've seen a few proofs of that online however they all kind of confuse me I was hoping somebody could show me the simplest proof there is.

I know we have to use the fact that the closure of S is equal to the union of S and all of its boundary points. Or that the closure of S is the union or S and the set of all its accumulation points.

Could somebody help me out

2 Answers 2

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I think an easy way to prove this is by contrapositive. If a set is not bounded, can you see why its closure must be unbounded?

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    is that because the set of boundary points would be empty therefore the the close of a set would be equal to the union of a set and its empty boundary point set so then the closure of a set would equal the set? – 2017-02-05
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    That proves the (obvious) fact that a set with a bounded closure is itself bounded. The proof by contradiction for the actual question would prove that if the closure is unbounded, then the set is unbounded, which is no easier than the direct proof. – 2017-02-05
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    The closure contains the original set, so if the original set isn't bounded... – 2017-02-05
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    @RobArthan yes I misread the question! – 2017-02-05
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    @pwerth: never mind, the OP doesn't seem to care! – 2017-02-05
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If the closure is not bounded , then take a natural number N which is more than the diameter of the original bounded set. Then we get a point x which is atleast at distance N+1 from another point y in the set. Now this point can't be in the original set , so it comes while taking the closure of some point z in the orginal set now we know $ d(x,z) + d(z,y) \ge d(x,y) $ which due to closure shows that it is maximum N which gives a contradiction.

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    N.b. the above is a direct proof that if $X$ has diameter bounded by $N$ then $\mathbf{cl}(X)$ has diameter bounded by $N + 1$. Assuming the closure is not bounded and then deriving a contradiction is an unnecessary logical detour. – 2017-02-05
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    yup .. you are right .. ok thanks for pointing it out . – 2017-02-05