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1. Order of congruence classes

Homework Statement If m\inZ and 2\leq n\in Z, then |[m]_n|=\frac{n}{(m,n)} Homework Equations Lagrange's Theorem The Attempt at a Solution I am confused simply because it seems like the problem might be missing something. We are asked to find the order of the congruence class m modulo n...
2. Countable union of countable sets vs countable product of countable sets

I know that a countable union of countable sets is countable, and that a finite product of countable sets is countable, but even a countably infinite product of countable sets may not be countable. Let X be a countable set. Then X^{n} is countable for each n \in N. Now it should also be true...
3. Is there a limit theorem for this?

Homework Statement The problem is much bigger, but part of the proof I've written hinges on the assumption that for a \in \mathbb{R}, x_n converging to x, a^{x_n} converges to a^x Homework Equations n/a The Attempt at a Solution I have tried taking $\log_a$ of both sides but that only...