It's very easy to prove that any convergent sequence is Cauchy, because, if $x = \lim x_n$, then $|x_n-x_m|<|x_n-x|+|x_m-x|$. Those cases don't give us more than than the standard limit definition.
The more interesting cases are when we don't immediately "know" what the limit is gonna be.
Let $0<\alpha<1$. If $f:\mathbb R\rightarrow \mathbb R$ (or $f:[a,b]\rightarrow [a,b]$ for some $a,b$,) such that $|f(x)-f(y)|<\alpha |x-y|$ for all $x,y$.
Given any $x_0$, define $x_{n+1}=f(x_n)$. Then $\{x_0,x_1,...,x_n,...\}$ is easily shown to be Cauchy, but it's much less obvious that it converges.
(The interesting side-affect of this example is the the limit $x=\lim x_n$ has the property that $f(x)=x$. But we can show that $f$ can only have one "fixed" value by our condition above. If $x\neq y$ and $f(x)=x$ and $f(y)=y$, then $|f(x)-f(y)|=|x-y|>\alpha |x-y|$.
For example, if $\beta$ is any real, then $f(x)=\beta+\alpha \arctan x$ has this property. It's not obvious what $\{x_0,f(x_0),f(f(x_0)),...\}$ converges to, but we know it converges.
Similarly, if $\{b_1,b_2,...,b_n,...\}$ is a convergent increasing sequence of real numbers, and $\{a_1,a_2,...,a_n,...\}$ has the property that for all $n$, $|a_{n+1}-a_n|\leq b_{n+1}-b_n$ then $\{a_n\}$ is easily shown to be Cauchy, even though we might have no idea what it converges to.