# How to show convergence in a metric space?

Suppose that $\{x_n\}→x$ where $\{x_n\}$ is a sequence in a normed space V and $x ∈ V$. Show that $\forall y ∈ V, \{x_n + y\} → x + y$.

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I think you should include the term "vector" somewhere in your post, because a metric space doesn't have to be a vector space – you Mar 6 '12 at 18:39

## 1 Answer

Recall the definition of convergence in a metric space: A sequence $\{x_n\}$ is said to converge to $x$ if $\forall\epsilon>0\ \exists N\in\mathbb{N}$ such that $n\geq N\implies d(x,x_n)<\epsilon$. For a normed vector space, $d(x,x_n)=|x-x_n|$

The result then follows from the definition and the fact that $|(x+y)-(x_n+y)|=|x-x_n|$

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Editted: Ah-hah, thank you! – D.G.S Mar 6 '12 at 18:49
That was nicely done. – ThisIsNotAnId Mar 6 '12 at 18:52
This is why knowing the basic inequalities of analysis-like Cauchy-Schwatz and the Triangle Inequality-is so important. – Mathemagician1234 Mar 6 '12 at 19:19
Although this particular analysis problem did not need the triangle inequality, most of them do :) – you Mar 6 '12 at 19:35