400: For Complete Metric Space, Closed Subspace Is Complete

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A description/proof of that for complete metric space, closed subspace is complete

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Topics

About: metric space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of complete metric space.
• The reader knows a definition of closed set.

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Target Context

• The reader will have a description and a proof of the proposition that for any complete metric space, any closed subspace is complete.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

For any complete metric space, $T_1$, any closed subspace, $T_2\subseteq T_1$, is complete.

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2: Proof

Let $p_1,p_2,...,$ where $p_i\in T_2$ be any Cauchy sequence on $T_2$. It converges on $T_1$ to a point, $p\in T_1$. Let us suppose that $p\notin T_2$. $p\in T_1\setminus T_2$ where $T_1\setminus T_2$ is open on $T_1$. There is an open ball, $B_{p-ϵ}\subseteq T_1\setminus T_2$, around $p$, and $p_i\notin B_{p-ϵ}$ for each $i$, which is a contradiction against $p$ is a convergent point of the sequence. So, $p\in T_2$. The sequence converges to $p$ on $T_2$, because for any open ball, $B_{p-ϵ}^{\prime }\subseteq T_2$, $B_{p-ϵ}^{\prime }=B_{p-ϵ}\cap T_2$, and there is an $i_0$ such that for any $i$ such that $i_0<i$, $p_i\in B_{p-ϵ}\cap T_2=B_{p-ϵ}^{\prime }$.

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References

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