Uniform Continuity and Cauchy Sequences

Question

Let $(S,d)$ and $(S^*, d^*)$ be metric spaces. If $f:S \to S^*$ is uniformly continuous and if $(s_n)$ is a Cauchy sequence in $S$, then $(f(s_n))$ is a Cauchy sequence in $S^*$.

$f:S \to S^*$ is uniformly continuous: $(\forall \varepsilon >0)( \exists > 0)(\forall s,t \in S)$ $d(s,t) < \delta \implies d(f(s), f(t)) < \varepsilon$.

$(s_n)$ Cauchy: $(\forall \varepsilon > 0)(\exists N \in \mathbb{N})$ such that $\forall n,m \geq N$ we have $d(s_n,s_m)) < \varepsilon$.

Now I need to show that $(\forall \varepsilon > 0)(\exists N \in \mathbb{N})$ such that $\forall n,m \geq N$ we have $d(f(s_n),f(s_m)) < \varepsilon$.

Since $f$ is uniformly continuous, $\delta$ will depend on $\varepsilon$ only. Then I separate the problem into two cases:

Suppose $\varepsilon < \delta$: Since I know $(s_n)$ is Cauchy, I can take the same $N$ to get $d(s_n,s_m) < \varepsilon < \delta$ which implies $d(f(s_n),f(s_m)) < \varepsilon$ by uniform continuity.

Suppose $\delta \leq \varepsilon$: Since $f$ is uniformly continuous, $d(s,t) < \delta \implies d(f(s),f(t)) < \varepsilon$ will hold for all $s,t \in S$. So let $N=1$, then $d(s_n,s_m) < \delta$ which implies $d(f(s_n),f(s_m)) < \varepsilon$.

Could someone give me feedback on my proof?

Edit For every $\varepsilon > 0$, since $f$ is uniformly continuous, there exists a $\delta >0$ such that for all $s, t \in S$, $d(s, t) < \delta \implies d(f(s),f(t)) < \varepsilon$. Then for any $\delta >0$ there exists an $N \in \mathbb{N}$ such that for all $n,m \geq N$, $d(s_n, s_m) < \delta$. By uniform continuity, $d(s_n, s_m) < \delta \implies d(f(s_n),f(s_m)) < \varepsilon$.

Answer 1

The first part (case $\varepsilon<\delta$) is correct, but I can't follow the second part. The conclusion $N=1$ is clearly wrong for arbitrarily small $\varepsilon$'s.

In the definition of uniform continuity, the presence of $\delta$ is always around $S$ and $\varepsilon$ is around $S^*$. So, we want to prove $f(s_n)$ is Cauchy:

Let's assume an $\varepsilon>0$ is given. For this we can choose a $\delta$, and for this $\varepsilon':=\delta$ we can choose an $N$ for $s_n$ by the Cauchy property.