Suppose f : (M, d)\rightarrow(N, p) is continuous. Prove or

estadsocm3

estadsocm3

Answered question

2022-01-22

Suppose f : (M,d)(N,p) is continuous.
Prove or disprove that f is uniformly continuous f maps a Cauchy seq. to Cauchy seq.

Answer & Explanation

Prince Huang

Prince Huang

Beginner2022-01-23Added 15 answers

We have fM,d(N,ρ)
Let ξ>0 and m,n be any two points. Since f is uniformly continuous, there exists δ>0, s.t. for every m,nM, distance between (m,n)<δdistance between [f(m),f(n)]<ξ.
Since xm is a cauchy sequence in M for every mM s.t. m,nn0
ρ[f(m),f(n)]<δ
ρ[f(xm),f(xn)]<ξ
Hence, image of a cauchy seq. under a uniform continuous function is again a cauchy sequence. (This property is not true foe arbitrary continuous function)

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