Convergence of finite metric spaces to an infinite one Let $\{(M_i, d_i)\}$ be an infinite sequence of finite metric spaces, where $|M_i|$ is strictly increasing with $i$.  Is there a standard definition of what it means for the sequence $\{(M_i, d_i)\}$ to converge to a (possibly infinite) metric space $(\bar{M}, \bar{d})$?
I am not entirely sure what "converges" should mean here, but I am interested in any reasonable notion that's been previously useful in math.  Perhaps a good definition is something like: for any $\epsilon > 0$, there is $I$ such that for all $i \ge I$, there is an embedding $\phi$ of $M_i$ into $\bar{M}$ such that (1) the set of $\epsilon$-balls centered at $\phi(m)$ for all $m \in M_i$ covers $\bar{M}$, and (2) for any $m_1, m_2 \in M$, we have $d_i(m_1, m_2) \in (\bar{d}(\phi(m_1), \phi(m_2)) - \epsilon, \bar{d}(\phi(m_1), \phi(m_2)) + \epsilon)$.
As a follow-up question: if there is a good definition of convergence along these lines, is there any sense in which the set of finite metric spaces with bounded diameter (max distance between two points) is compact (i.e. any infinite sequence contains a convergent subsequence)?
 A: There is indeed a standard definition like this, known as Gromov-Hausdorff convergence.  You can learn much more by looking up this term; let me just briefly state the definition.  First, if $A$ and $B$ are nonempty subsets of a metric space $X$, define $d_H(A,B)=\max(\sup_{a\in A} d(a,B),\sup_{b\in B}d(b,A))$ (this is called the "Hausdorff metric" on subsets of $X$).  Now if $M$ and $N$ are two nonempty metric spaces, define $d(M,N)$ to be the infimum of $d_H(f(M),g(N))$ over all isometric embeddings $f:M\to X$ and $g:N\to X$ of $M$ and $N$ into another metric space $X$.  This infimum is finite as long as $M$ and $N$ are bounded, and defines a metric on the set of all nonempty compact metric spaces up to isometry (if you allow noncompact spaces, it is possible to have $d(M,N)=0$ without $M$ and $N$ being isometric).
On your follow-up question, no, the set of finite metric spaces with bounded diameter is not compact.  In fact, the set of finite metric spaces of diameter $\leq R$ is dense in the set of all compact metric spaces of diameter $\leq R$.  This latter space is still not compact.  For a simple example, consider the case that $M_i$ has $i$ points which all have distance $1$ from each other.  No subsequence of this can Gromov-Hausdorff converge to a metric space (even a non-compact space), essentially because any limit would have to be an infinite space in which the distance between any two points is $1$, but finite subsets of such a space do not converge to the whole space with respect to the Hausdorff metric (since the distance between any finite subset and the whole space will be $1$). 
