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Interweaving sequences – Some Mathematics

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Here is a mildly interesting fragment of mathematical reasoning which can be occasionally be applied to proofs in topology and analysis. I illustrate it with two examples and give no additional explanation. Proposition Let ( 𝑀 , 𝑑 ) , ( 𝑁 , 𝑑 ′ ) (M,d),(N,d ′ ) be metric spaces, with dense subsets 𝐴 , 𝐵 ⊆ 𝑀 , 𝑁 A,B⊆M,N respectively. Further suppose that 𝑁 N is complete. Then, if 𝑓 : 𝐴 → 𝐵 f:A→B is uniformly continuous, there exists a unique continuous extension 𝑓 ^ : 𝑀 → 𝑁 f ^ ​ :M→N such that 𝑓 ^ ∣ 𝐴 = 𝑓 f ^ ​ ∣ A ​ =f. Proof Define 𝑓 ^ : 𝑀 → 𝑁 f ^ ​ :M→N, by: There are a number of items to check here. Firstly, we know that there does exist a sequence ( 𝑎 𝑛 ) ∈ 𝐴 , 𝑎 𝑛 → 𝑥 (a n ​ )∈A,a n ​ →x since 𝐴 A is dense. Furthermore, we have: ( 𝑎 𝑛 ) (a n ​ ) convergent ⟹ ( 𝑎 𝑛 ) ⟹(a n ​ ) Cauchy ⟹ ( 𝑓 ( 𝑎 𝑛 ) ) ⟹(f(a n ​ )) Cauchy (by uniform continuity) ⟹ ( 𝑓 ( 𝑎 𝑛 ) ) ⟹(f(a n ​ )) convergent (by the completeness of 𝑁 N). T

Here is a mildly interesting fragment of mathematical reasoning which can be occasionally be applied to proofs in topology and analysis. I illustrate it with two examples and give no additional explanation. Proposition Let (M,d),(N,d') be metric spaces, with dense subsets A, B \subseteq M,N respectively. Further suppose that N is complete. Then, if f: A \to B is uniformly continuous, there exists a unique continuous extension \hat{f}: M \to N such that \hat{f} \vert_A = f . Proof Define \hat{f}: M \to N , by: \hat{f}(x)=\lim_{a\in A, a \to x}f(a) There are a number of items to check here. Firs

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