Let \begin{aligned} t_{k}=\dfrac{kT}{n} ,\ \ k\in \left[ 0,n\right]\\ \sigma(x)=\sigma\ constant \ and \ \sigma>0\\ b\ and \ b'\ lipschitz \ continuous \\ for \ s \ in \ [t_k,t_{k+1})\ ,\underline{s}=t_k \end{aligned}
and a diffusion and its associated continuous Euler Scheme
\begin{aligned} X_{t}=X_{0}+\int ^{t}_{0}b\left( X_{s}\right) ds+\int ^{t}_{0}\sigma dW_{s}\\ \overline{X}_{t}=X_{0}+\int ^{t}_{0}b\left( \overline{X}_{\underline{s}}\right)ds +\int ^{t}_{0}\sigma dW_{s}\\ \end{aligned}
I must prove prove that
\begin{aligned} \sup_{0 \leq t \leq T}\left| \int ^{t}_{0}\left( b\left( \overline{X}_{s}\right) -b\left( \overline{X}_{\underline{s}}\right) \right) ds\right|\leq\left[ b\right] _{Lip}\dfrac{T}{n}\int ^{T}_{0}\left| b\left( \overline{X}_{\underline{s}}\right) \right| ds+\left[ b'\right] _{Lip}\int ^{T}_{0}\left| \overline{X}_{s}-\overline{X}_{\underline{s}}\right| ^{2}ds+\sigma\sup_{0 \leq t \leq T}\left|\int ^{t}_{0}b'\left( \overline{X}_{\underline{s}}\right) \left( W_{s}-W_{\underline{s}}\right) ds\right| \end{aligned}
I tried to use Ito's Lemma with b but I'm not sure that I have the correct hypothesis and I still cannot get the above inequality, only some terms(first and last one). I cannot futher transform the remaining term to make the second one appear