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Prove the IVT theorem Prove: If f is continuous on [a,b] and f(a),f(b) have different signs...

Prove the IVT theorem

Prove: If f is continuous on [a,b] and f(a),f(b) have different signs then there is an r ∈ (a,b) such that f(r) = 0.

Using the claims:

f is continuous on [a,b]

there exists a left sequence (a_n) that is increasing and bounded and converges to r, and left decreasing sequence and bounded (b_n)=r.

limf(a_n)= r= limf(b_n), and f(r)=0.

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