[Math] If $f$ is an integrable function on $[a,b]$, then $f$ is integrable on $[c,d]$ $\forall [c,d]\subseteq [a,b]$

Question

If $f$ is an integrable function on $[a,b]$, then $f$ is integrable on $[c,d]$ $\forall [c,d]\subseteq [a,b]$

My solution:

Let $P'=\{a,c,d,b\}$ be a partition of $[a,b]$. Let $P$ be a refinement partition of $P$ such that $||P||<\delta$

$f$ is integrable on $[a,b]\implies \exists P\text{ of } [a,b] \text{ s.t. }\forall \epsilon>0~ S(P)-s(P)\leq \epsilon $ where $S(P),s(P)$ is the upper and lower Riemann sums over the partion P.

$$\sum^n_{i=0}\left(M_i-m_i\right)\left(x_i-x_{i-1}\right)\leq \epsilon$$
Suppose at $i=k$ we have $(c,x_{k-1})$ and at $i=k+m$ we have $(x_{k+m},d)$, Since each term in the summation is nonnegative then:
$$\sum^{k+m}_{i=k}\left(M_i-m_i\right)\left(x_i-x_{i-1}\right)\leq \sum^{k}_{i=0}\left(M_i-m_i\right)\left(x_i-x_{i-1}\right)+\sum^{k+m}_{i=k}\left(M_i-m_i\right)\left(x_i-x_{i-1}\right)+\sum_{k+m+1}^{n}\left(M_i-m_i\right)\left(x_i-x_{i-1}\right)\leq \epsilon$$
$$\implies\text{ f is integrable on }[c,d]$$

Is my solution correct?

Answer 1

You wanting to prove that $f$ is integrable of $[c,d]$, i.e.

For every $\epsilon>0$ there exists $\delta>0$ such that for every partition $P$ of $[c,d]$ that is finer than $\delta$, we have $S(P)-s(P)<\epsilon$.

So start with "Let $\epsilon>0$ be given." Then use integrability of $f$ on $[a,b]$ to obtain some suitable $\delta$ (that may depend on $\epsilon$). Then assume that $P$ is any partition of $[c,d]$ that is finer than $\delta$. From this, construct a suitable partition $P'$ of $[a,b]$ that allows you to bound $S(P)-s(P)$ (presumably in terms of $S(P')-s(P')$).