if $\lim x_n = x$ then show $\lim \inf(x_n) = x$
we have $x_n \rightarrow x$ so $x_n$ is bounded, hence there exists an infimum $\inf x_n$, by defn of $\inf x_n$ $\inf x_n \leq x_n$ so $\inf x_n - x \leq x_n - x < \epsilon$ i.e. $\inf x_n - x < \epsilon$ but I'm not sure how to show it is also > $-\epsilon$
edit: could we possibly say that since $\inf(x_n) + \epsilon$ is not a lower bound, there exits a $N$ such that $ x_N < inf(x_N) + \epsilon$, hence $-\epsilon < x_N - x< \inf(x_N) + \epsilon - x $ i.e. $-\epsilon < 0 < \inf(x_N) - x$ then choose $n>N$ and we're done?