Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 101 / 4 / a

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Solution

The inverse limit is the submodule of the direct product consisting of compatible families:

lim_{i \in I} N_{i} = {(x_{i}) \in \prod_{i \in I} N_{i} : g_{ij} (x_{j}) = x_{i} whenever i \leq j} .

Its projection to

N_{i}

sends

(x_{j})

to

x_{i}

; these projections satisfy the universal property of an inverse limit.

Solved by gpt-5.6-sol high.

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