Hey Guys I'm studying this proof in real analysis and just this one line gets me:
Suppose that f is bounded and Riemann integrable on [a,b] and ε > 0 is given. Since L = sup{L(P): P is a partition on [a,b]} exists, there is a partition P1 of [a,b] such that
L(P1) > L - (1/2)ε
What exactly does this statement mean? and where does the 1/2 come from? I'm so confused
It is just the definition of supreme. If $L<\infty$ is the supremum of a set $A$ of real numbers then
$$\forall \epsilon >0 \exists a\in A : a>L-\epsilon.$$
Since $\epsilon$ is arbitrary the result holds for $1/2\epsilon.$
Probably you have encountered this in the middle of a proof and the reason for $1/2\epsilon$ is technical. Sure there is another term in the proof with the same $1/2\epsilon.$ Thus, when you add both terms you get the "familiar" $\epsilon.$