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?