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,y∈E 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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