I got stuck on proving this exercise for quite a while and found the proof of this statement Positive operator is bounded :
For a real Banach space $E$ let $T:E\rightarrow E'$ be a positive operator in the sense that $(Tx)(x)\geq 0$ for all $x\in E$. Show that $T$ is bounded.
My question is, how exactly did we even come up with this proof construction at the first place? What I did was starting with $x_n \to x_0 \in E$ and $T(x_n) \to f \in E'$ and attempt to show $T(x_0) = f$. Namely, $T(x_0)(y) = f(y)$ for all $y \in E$. And then I literally just got stuck. The relation of the positiveness identity does not seem to "naturally" arise from this point (except from I could recognize that this might help us to show two-sided inequality). But how could we proceed beyond this point naturally? Or should I just take this proof as one of the proofs that does not have intuition at all and move on.