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## Homework Statement

Suppose that f has the intermediate value property on an interval J, that g has the intermediate value property on an interval I and that g(I) is a subset of J. Prove that f°g has intermediate value property on I.

## Homework Equations

## The Attempt at a Solution

I think I might have simplified too much and missed the point. Here is my work so far...

Since f has IVP on J, there is an a,b in J and a≠b and v is a number between f(a) and f(b) such that there is a c between a and b that f(c)=v.

Similarly for g, there is a e,f in I and e≠f and u is a number between g(e) and g(f) such that there is a d between e and f that g(d)=u.

Since g(I) is a subset of J, f(g(I)) is a subset of f(J).

Then f(g(e)) and f(g(f)) are numbers such that f(g(e))≠f(g(f)). Since g(d)=u is between e and f, then f(u) is between f(g(e) and f(g(f)) since f has IVP on J.

∴f°g has IVP on I.