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This is a question about integration, which I've just started studying in my real analysis course.

Terminology: Let $P$ be a partition of the interval $[a, b]$. Let $f$ be a function on $[a, b]$. Let $L(f, P)$ be the sum of the areas of rectangles whose height is determined by the minimum of $f(x)$ in a particular subinterval. Similarly, let $U(f, P)$ be the sum of the areas of rectangles whose height is determined by the maximum of $f(x)$ in a particular subinterval.

Let $P_1$ and $P_2$ be any two partitions of $[a, b]$. In my prof's lecture slides, a theorem called "the partition theorem" states that if $f$ is bounded on $[a, b]$ then $L(f, P_1)\le U(f, P_2)$.

My prof's lecture slide then says:

Important inferences that follow from the partition theorem:

For any partition $P$', the upper sum $U(f, P')$ is an upper bound for the set of all lower sums $L(f, P)$.

So far, so good.

It then says:

$\therefore\sup\{L(f, P)\text{ : $P$ a partition of }[a, b]\}\le U(f, P')\ \forall P'$

How did he get that?

The slide then says:

$\sup\{L(f, P)\}\le\inf\{U(f, P)\}$

How did he get that?

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  • $\begingroup$ If something is an upper bound, then it must be at least as big as the supremum. Otherwise consider values strictly above the upper bound and strictly below the supremum: are these upper bounds or not? (Or short circuit this by using a definition of supremum as least upper bound) $\endgroup$ – Henry Nov 27 '16 at 0:41

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