Tuesday, 18 March 2014

real analysis - Continuous mapping of a compact metric space is uniformly continuous

Theorem: Let f be a continuous mapping of a compact metric space X into a metric space Y. Then f is uniformly continuous on X.




Proof: Let f is not uniformly continuous then for some ε>0 and arbitrary δ>0 exists x,yE and dX(x,y)<δ but dY(f(x),f(y))>ε. Taking δ=1n then exists sequences {pn},{qn} in X such that dX(pn,qn)0 but dY(f(pn),f(qn))>ε.



Since X is compact then {pn} and {qn} has limit point, namely p and q. Then exists some subsequence {pnk} that converges to p. Using triangle inequality we get that {qnk} also converges to p.



Since f is a continuous mapping then f(pnk)f(p) and f(qnk)f(p). Combine these statements we get $\varepsilon

Is this proof correct?

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