# Tag Info

### Is an infinite composition of bijections always a bijection?

We shouldn't expect limits (of any sequence of functions) to preserve injectivity, since limits can change strict inequalities into nonstrict inequalities. For a concrete counterexample, consider each ...
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### Is an infinite composition of bijections always a bijection?

There's a very simple example, actually. Take $f_i (x) = \sqrt{x} = x^{1/2}.$ Over $X = (0, \infty),$ for each $i \in [1..]$, $f_i$ is bijective, meeting the requirements of the question. With this ...

### Is an infinite composition of bijections always a bijection?

Here is a countable counterexample. Consider the one-point compactification $X = \mathbb{N} \cup \{ \infty \}$ of $\mathbb{N}$ and the sequence of bijections $f_i : X \to X$ given by the ...
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### Why is $\frac{p}{-q}$ written as $\frac{-p}{q}$ and not as $\frac{p}{-q}$?
Regarding notation, I'm fond of a modified Einstein index contraction notation: $f^0_n : X_0 \to X_n$ where $f^0_n = f_n \circ \cdots \circ f_1$. It's nice because then you also have notation for ...