[aumaim]

Ex. Suppose that f is integrable on [a,b] and that [c,d]$\subseteq$ [a,b].Prove that f is integrable on [c,d].

Proof. We will show that f is integrable on [c,d] ,
we have f is integrable on [a,b] by theorem 26.9
f is integrable on [a,b]$\Leftrightarrow \forall \varepsilon $>0,$\exists$ a partition P of [a,b] s.t.
U(f,p) - L(f,p)<$\varepsilon$
and that [c,d]$\subseteq $[a,b].Then f is integrable on [c,d].
Let $\varepsilon $>0.Thus there exists a partition P of [c,d] such that
U(f,p) - L(f,p) < $\varepsilon $
Hence f is integrable on [c,d]


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