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## Homework Statement

Sets A and B are sets of positive real numbers. Define C = {st| s \in S and t \in T}

Prove inf(C) = inf(S)*inf(T)

## The Attempt at a Solution

so i'm trying to prove inf(C) <= inf(S)*inf(T) and inf(C) >= inf(S)*inf(T).

i'll use e as epsilon. epsilon is positive

By definition there is an s in S such that: s < inf S + e. There is a t in T such that: t < inf T + e. Additionally, inf(C) <= st for all s in S and t in T by definition.

st < (inf S + e)(inf T + e) = (inf S)(inf T) + (inf S)*e + (inf T)*e + e^2

inf C <= st <= (inf S)(inf T) < inf S)(inf T) + (inf S)*e + (inf T)*e + e^2

(NOTE: i'm not sure about the middle inequality: "st <= (inf S)(inf T)" )

is this a correct way to go about this? I'm also not sure about proving the other direction