Suppose that $\{x_n\}→x$ where $\{x_n\}$ is a sequence in a normed space V and $x ∈ V$. Show that $\forall y ∈ V, \{x_n + y\} → x + y$.
How to show convergence in a metric space?
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metric-spaces
normed-spaces
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1I think you should include the term "vector" somewhere in your post, because a metric space doesn't have to be a vector space – 2012-03-06
1 Answers
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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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0Although this particular analysis problem did not need the triangle inequality, most o$f$ t$h$em do :) – 2012-03-06