Relation between riemann sums of 3 functions

Question

I have a doubt in this multiple select question of Riemann Sum.

It says that if $f,g,h$ are bounded functions on [$a,b$]. Let $f$($x$) $\le$ $g$($x$) $\le$ $h$($x$) for all x in[a,b]. Then which of the following is always true?

1. If $U$($h$,$P$)-$U$($f$,$P$) $\lt$ $1$ then $U$($g$,$P$)-$L$($g$,$P$) $\lt$ $1$
2. If $U$($h$,$P$)-$L$($f$,$P$) $\lt$ $1$ then $U$($g$,$P$)-$L$($g$,$P$) $\lt$ $1$
3. If $U$($h$,$P$)-$L$($h$,$P$) $\lt$ $1$ then $U$($g$,$P$)-$L$($g$,$P$) $\lt$ $1$
4. If $L$($h$,$P$)-$U$($f$,$P$) $\lt$ $1$ then $U$($g$,$P$)-$L$($g$,$P$) $\lt$ $1$

I have proven that option $2$ is always correct. But I am unable to check the remaining options. I can't find the examples to discard them.

Answer 1

Counterexamples for $(1),(3),(4)$ . . .

Let $a=0,b=1$, and let $P$ be the crude partition with just one interval.

For $(1)$, let $f(x)={\large{\frac{3}{4}}}x,\;\,g(x)=x,\;\, h(x)={\large{\frac{5}{4}}}x$.

For $(3)$, let $f(x)=x,\;\,g(x)=x,\;\,h(x)=1$.

For $(4)$, let $f(x)=x,\;\,g(x)=x,\;\,h(x)=x$.