Given $X(0) \in \Bbb R$ and a continuous and bounded real-valued function $g(.)$ from $\Bbb R \to \Bbb R$, for $\delta > 0$ define the sequence $\{X_n^\delta\}$ by $\{X_0^\delta\} = X(0)$ and
$$ X_{n+1}^\delta = X_{n}^{\delta} + \delta g(X_{n}^{\delta}), n \geq 0$$
Define the piecewise linear interpolation $X^{\delta}(.)$ by
$$X^{\delta}(t) = \frac{t-n\delta}{\delta} X_{n+1}^\delta + \frac{n\delta +\delta -t}{\delta} X_{n}^\delta \text{ on }[n\delta, n\delta + \delta).$$
Then I need to prove that $$ X^{\delta}(t) = X(0) + \int_{0}^{t} g(X^{\delta}(s))ds + \rho^{\delta}(t)$$
where the interpolation error goes to zero as $\delta\to 0$.
I am not much aware of such things. Any hint will be helpful.