Here is a useful limit theorem which can be used to prove many of the commonly needed limit theorems as special cases.
Let , , , and be topological spaces, where is a subspace of . Let be such that , where and . Let be continuous at . Then
Let be such that , and be continuous at . Then
Actually, this is equivalent to continuity at .
Let , where is a subspace of . Let be a continuous function. Then
Let be such that and be such that . Since the projection is continuous,
Actually, the reverse implication also holds: if the limits of all component functions exist, then so does the limit of the function (with the relation above).
Let and be topological semigroups (e.g vector spaces), , and be such that . Since addition is continuous,
We also used the result of example 3 here.
Let , , and be topological -modules (e.g. vector spaces), , and be such that , where . Since by definition the multiplication is continuous, so is its restriction to a fixed , and thus
Denote by the neighborhoods of a point (and similarly for other spaces). Let , and . Since is continuous at , there exists such that . By the definition of a limit of a function, there exists such that . Therefore , i.e. . QED.