We know that a function $f$ is continous iff for every convergent sequence $\{S_n\}$, $f(S_n)$ is convergent.
Likewise, I just encountered a theorem that states $f$ is uniformly continuous if for every Cauchy sequence $\{S_n\}$, $f(S_n)$ is Cauchy. (I believe the converse is true?)
But what I don't understand is that uniform continuity is stronger than continuity, but being Cauchy is weaker than being convergent. Thus, why can't we write the theorem on uniformly continuous functions in terms of convergent sequences. In other words, if $uniform$ $continuity \Rightarrow continuity$ then why don't we have $uniform$ $contininuity$ $\Leftarrow\Rightarrow$ it maps convergent sequences to convergent sequences?
Best Answer
Any continuous function maps convergent sequences to convergent sequences and continuous functions need not be uniformly continuous. So your equivalence does not hold. (I think the mistake you are making is to think only in one direction and then make an iff statement).