Find $\lim\_{k \rightarrow \infty} \sup a\_k$ and $\lim\_{k \rightarrow \infty} \inf (a\_k)$, with $a\_k=(\frac{1}{k})$

Per definition: $\lim\_{k \rightarrow \infty} \sup(a\_k) = \lim\_{k \rightarrow \infty} \sup (\bigcup\_{i=k}^\infty \{a\_i\})$

I have $\sup (\bigcup\_{i=k}^\infty \{a\_i\}) = \frac{1}{k}$

$\Rightarrow \lim\_{k \rightarrow \infty} \sup (\bigcup\_{i=k}^\infty \{a\_i\}) = 0$

For the limes inferior I get:

$\lim\_{k \rightarrow \infty} \inf(a\_k) = \lim\_{k \rightarrow \infty} \inf (\bigcup\_{i=k}^\infty \{a\_i\})$

Since $\inf (\bigcup\_{i=k}^\infty \{a\_i\}) = 0$

$\Rightarrow \lim\_{k \rightarrow \infty} \inf(a\_k) = 0$

Is that correct? Can it be that the limes superior and limes inferior of a sequence are equal?