Let $(z\_n) \subset \mathbb{R}$ be a sequence defined, for each $n \in \mathbb{N}$, by:

$$z\_n=min\{x\_n,y\_n\}$$

With $(x\_n)$ and $(y\_n)$ being two convergent sequences in $\mathbb{R}$ with $\lim{x\_n}=a$ and $\lim{y\_n}=b$.

Show that $(z\_n)$ is also convergent and calculate the limit.

First I've separated in two cases: $a=b$ and $a \not= b$.
For the first one, we have:

* $\forall \varepsilon > 0, \exists n\_0 \in \mathbb{N} ; n > n\_0 \Rightarrow |x\_n - a|<\varepsilon$
* $\forall \varepsilon > 0, \exists n\_1 \in \mathbb{N} ; n > n\_1 \Rightarrow |y\_n - a|<\varepsilon$

So, if $n\_\delta = max\{n\_0,n\_1\}$, we have:
$$n > n\_\delta \Rightarrow |z\_n-a|<\varepsilon$$
By the defnition of $z\_n$ sequence. (Right?)

But what about the second case? I can't come up with anything. Leads?