Let f , g and h be bounded functions on the closed interval [a,b] such that $f(x)\leq g(x)\leq h(x)\;\forall x\in[a,b]$. Let $P=\{a=a_0<a_1<a_2.....<a_n=b\}$ be the partition of [a,b], We denote by U(f,P) and L(f,P), the upper and lower Riemann sums of f with respect to the partition P similarly g and h.Which one is necessarily true.
A. $ If\;U(h,P)-U(f,P)<1\;then\;U(g,P)-L(g,P)<1$
B. $If\; L(h,P)-L(f,P)<1\;then\;U(g,P)-L(g,P)<1$
C. $ If\;U(h,P)-L(f,P)<1\;then\;U(g,P)-L(g,P)<1$
D. $If\;L(h,P)-U(f,P)<1\;then\;U(g,P)-L(g,P)<1$
Let $P=\{x_0<x_1<...<x_n\}$
$U(g,P)=\sum _{i=1}^n M_i\Delta x_i$;$U(h,P)=\sum _{i=1}^n M_i^{'}\Delta x_i$;
$U(f,P)=\sum _{i=1}^n M_i^{''}\Delta x_i$;$M_i=\sup _{[x_{i-1},x_i]}g(x)$;$M_i^{'}=\sup _{[x_{i-1},x_i]}h(x)$;$M_i^{''}=\sup _{[x_{i-1},x_i]}f(x)$
Since $f(x)\leq g(x)\leq h(x) \forall x $ so we have $M_i^{'}\geq M_i\geq M_i{''}$
Similarly define for $L(f,P)$ and thus we obtain
$U(g,P)\leq U(h,P)$ and $L(f,P)\leq L(g,P)\implies -L(f,P)\geq -L(g,P)$.
Then $U(g,P)-L(g,P)\leq U(h,P)-L(f,P)<1$.
Hence $(c)$ is correct.