The following fact has been called "Ramsey's Theorem for Analysts" by H. P. Rosenthal.
Theorem. Let $(a_{i,j})_{i,j=0}^\infty$ be an infinite matrix of real numbers such that $a_i = \lim_{j\to\infty} a_{i,j}$ exists for each $i$ and $a = \lim_{i\to\infty} a_i$ exists too. Then there is an infinite sequence $k(0) < k(1) < k(2) < \cdots$ such that $a = \lim_{i<j} a_{k(i),k(j)}$.
The last limit means that for every $\varepsilon > 0$ there is an $n$ such that $n < i < j$ implies $|a-a_{k(i),k(j)}| < \varepsilon$. When the matrix is symmetric, this is just an ordinary double limit.
The proof is a straightforward applications of the two-dimensional Ramsey's Theorem. The obvious higher dimensional generalizations are also true and they can be established in the same way using the corresponding higher dimensional Ramsey's Theorem. These are used to construct "spreading models" in Banach Space Theory.