STATEMENT: Let $[a,b]$ be a finite interval in $\mathbb{R}$. Let $\Lambda$ be the collection of all finite subsets of $[a,b]$ that contain $a$ and $b$. When orderedf by inclusion, it is a directed set. For any $\lambda\in \Lambda$ we index its elements in increasing order, so $$\lambda=\left\{a=t_0<t_1<\cdots<t_n=b\right\}$$ for some $n$ depending on $\lambda$. Let $f$ be a bounded $\mathbb{R}$-valued function on $[a,b]$. For any $\lambda$, indexed as above, set $$S^\lambda_k(f)=\sup\left\{f(t):t_{k-1}\leq t\leq t_k\right\}$$ for $k=1,\cdots,n$.Define a net $\left\{I_\lambda^*(f)\right\}_{\lambda\in\Lambda}$ in $\mathbb{R}$ by $$I_\lambda^*(f)=\sum^n_{k=1}S^\lambda_k(f)(t_k-t_{k-1})$$
a)Prove that the net $\left\{I_\lambda^*(f)\right\}_{\lambda\in\Lambda}$ converges in $\mathbb{R}$.
QUESTION: I would like some hints as to how to begin such a problem(still learning about nets). Any help would be greatly appreciated.
PROOF: Lemma (Monotone Convergence Theorem):Given any net $\left\{x_λ\right\}_{λ∈Λ}$ in $R$ that is bounded and eventually monotonic, $\left\{x_λ\right\}_{λ∈Λ}$ converges.
Proof: Suppose we have a a net $\left\{x_λ\right\}_{λ∈Λ}$ in $\mathbb{R}$ such that if $θ≥λ$ then $x_θ≥x_λ.$ So it is a monotonically increasing net. Let $M∈R$, and assume that $|x_λ|<M$ for all $λ$ in our indexing set, we take $s=\sup\left\{x_λ\right\}_{λ∈Λ}$. We show that $\left\{x_λ\right\}_{λ∈Λ}$ converges to s. By definition of the supremum, given any open ball,$ U$, around s we can find an $x_λ$ in our net so that $x_λ∈U$. But note that for all $θ>λ, s>x_θ≥x_λ$. Thus for this chosen $λ$, we see that $x_θ∈U$. Hence our net converges to $s. $
The proof when $\left\{x_λ\right\}_{λ∈Λ}$ is a monotonically increasing net follows a similar proof and will be omitted.
To prove our original assertion, we first notice that $|f(x)|<M$ for some $M∈R$, since $f$ is bounded. Furthermore, when $λ_1≥λ→I_{λ_1}^∗<I_{λ}^∗.$ So our net is monotonically decreasing. By our lemma we see that our net converges to $\sup\left\{x_λ\right\}_{λ∈Λ}.$