(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

let {x_{i}} be a sequence of distinct elements in a metric space, and suppose that x_{i}→x. Let f be a one-to-one map of the set of x_{i}s into itself. prove that f(x_{i})→x

2. Relevant equations

by convergence of x_{i}, i know that for all ε>0, there exists some n_{0}such that if i≥n_{0}, then d(x_{i},x)<ε.

by one to one, i know that if f(x)=f(y), then x=y.

x_{i}s are distinct (which implies that there are an infinite number of points?)

f(x_{i}) ⊆ {x_{i}} (with this and one to one, does that imply the function is onto? i think yes, since the cardinality of the two sets are equal, but i'm not sure this will help me)

i want to show: for all ε>0, there exists some n_{0}such that if i≥n_{0}, then d(f(x_{i}),x)<ε.

3. The attempt at a solution

so i've been staring at this problem for hours, not getting farther than stating my assumptions. i'm having a hard time even convincing myself that it's true, which is usually my first step. i'm thinking that since there are an infinite amount of points, then no matter what function i have, any mapping will "fill up" the earlier holes, leaving me with a remaining series that lies in any epsilon neighborhood, but i'm having trouble expressing that mathematically, and i'm not even sure if it's true. i got my hopes up by trying use a contradiction and assuming the negation of what i'm trying to show, thinking that since f(x_{a})=x_{b}for some a, b in N, but all that does is show that there is a sequence, not one necessarily related by the i's (am i making sense? i mean i need the f(x_{i}) to converge, not create a new sequence by reordering my f(x_{i}) such that it converges), and we already knew that sequence exists.

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# Metric spaces and convergent sequences

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