Originally, I encountered this question on Terence Tao's blog, where the following exercise is presented:
Exercise 23 (Implications and equivalences) Let $X_n, X$ be random variables taking values in a $\sigma$compact metric space $R$.
[...]
(ii) Show that if $X_n$ converges in distribution to $X$, then $X_n$ has a tight sequence of distributions.
(iii) Show that if $X_n$ converges in probability to $X$, then $X_n$ converges in distribution to $X$. (Hint: first show tightness, then use the fact that on compact sets, continuous functions are uniformly continuous.)
Whats struck me as odd was that $R$ is merely assumed to be $\sigma$-compact, i.e. no completeness-assumption on $R$ is made (as the example $\Bbb{Q}$ shows, there are $\sigma$-compact spaces that are neither locally compact, nor complete).
This makes it rather hard to construct new compact subsets (from old ones). Indeed, the proofs of the above statement (ii) that I found (see e.g. https://www.math.leidenuniv.nl/~vangaans/jancol1.pdf Theorem 5.2) use the fact that
$$ K := \bigcap_j \bigcup_{i=1}^{k_j} \overline{B}(a_i, 1/j) $$ is a compact set if $R$ is complete, because it is closed and totally bounded.
Nevertheless, Wikipedia (http://en.wikipedia.org/wiki/Prokhorov%27s_theorem) does also not assume that the metric space in question is complete, there the only requirement is separability.
Note that we need to produce compact subsets of $R$, because for tightness (see below), we have to show that for $\varepsilon > 0$ there is $K \subset R$ compact such that $\Bbb{P}(X_n \in K) \geq 1-\varepsilon$ holds for all $n$ sufficiently large.
In summary, my question is if the two statements as in the exercise above are correct even without further completeness assumptions on $R$. Hints/proofs/counterexamples would be highly appreciated.
For convenience of the reader, I repeat the necessary definitions below:
Definition 10 (Modes of convergence) Let $R = (R,d)$ be a $\sigma$-compact metric space (with the Borel $\sigma$-algebra), and let $X_n$ be a sequence of random variables taking values in $R$. Let $X$ be another random variable taking values in $R$.
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$X_n$ converges in probability to $X$ if, for every $\epsilon > 0$, one has $$\liminf_{n \rightarrow \infty} {\bf P}( d(X_n,X) \leq \epsilon ) = 1$$ [...]
$X_n$ converges in distribution to $X$ if, for every bounded continuous function $F: R \rightarrow {\bf R}$, one has $$\lim_{n \rightarrow\infty} \mathop{\bf E} F(X_n) = \mathop{\bf E} F(X)$$
$X_n$ has a tight sequence of distributions if, for every $\epsilon > 0$, there exists a compact subset $K$ of $R$ such that $\mathop{\bf P}( X_n \in K ) \geq 1 - \epsilon$ for all sufficiently large $n$.