Sequence approaching infimum Given the fact that for a non-empty subset $K ⊂ V$ and $J : V → \mathbb R$ the number $\inf_{v \in K} J(v)$ exists, why does a minimizing sequence always exist?  So I read somewhere this: A minimizing sequence of a function $J$ on the set $K$ is a sequence $(u_n)_{n∈ \mathbb N}$ such that $u_n \in K$ for all $n$ and $\lim_{n\to\infty}J(u_n) = \inf_{v∈K} J(v)$. By deﬁnition of the inﬁmum value of $J$ on $K$ there always exist minimizing sequences! It seems like a logical consequence but I don't get it... 

Hint: 

Gap Lemma: Let $\varnothing \neq A \subseteq \mathbb R$ and $x = \inf A$. Given any $\varepsilon > 0$, there is an $a \in A$ such that $a - x < \varepsilon$.

Then, we can use the gap lemma with $\varepsilon_n = \frac{1}{n}$ to get a sequence $(a_n)_{n \in \mathbb N}$ such that $\lim_{n\to\infty} a_n = x$.
