[proofplan] The hypothesis says that the difference between the augmented statistic and the corresponding Dickey-Fuller statistic converges to zero in probability. The Dickey-Fuller statistic is assumed to converge in distribution to $L_D$. Slutsky's theorem then gives convergence in distribution of the sum $\tau_T^{\mathrm{DF}}(D)+(\tau_T^{\mathrm{ADF}}(D)-\tau_T^{\mathrm{DF}}(D))$ to $L_D+0=L_D$. [/proofplan]

[step:Decompose the augmented statistic into a known limit plus a negligible error]For each $T$, write \begin{align*} R_T=\tau_T^{\mathrm{ADF}}(D)-\tau_T^{\mathrm{DF}}(D). \end{align*} By assumption, \begin{align*} R_T\xrightarrow{p}0. \end{align*} Also by assumption, \begin{align*} \tau_T^{\mathrm{DF}}(D)\Rightarrow L_D. \end{align*} The augmented statistic can therefore be written as \begin{align*} \tau_T^{\mathrm{ADF}}(D)=\tau_T^{\mathrm{DF}}(D)+R_T. \end{align*}[/step]

[guided]The statement separates the hard time-series work from the limiting-distribution transfer. The hard work is the assumption \begin{align*} \tau_T^{\mathrm{ADF}}(D)-\tau_T^{\mathrm{DF}}(D)\xrightarrow{p}0, \end{align*} which says that the augmentation changes the statistic by an asymptotically negligible amount. Naming this difference as \begin{align*} R_T=\tau_T^{\mathrm{ADF}}(D)-\tau_T^{\mathrm{DF}}(D) \end{align*} gives $R_T\xrightarrow{p}0$. The other input is \begin{align*} \tau_T^{\mathrm{DF}}(D)\Rightarrow L_D. \end{align*} Since \begin{align*} \tau_T^{\mathrm{ADF}}(D)=\tau_T^{\mathrm{DF}}(D)+R_T, \end{align*} the augmented statistic is the Dickey-Fuller statistic plus a term vanishing in probability.[/guided]

[step:Apply Slutsky's theorem] Slutsky's theorem says that if $A_T\Rightarrow A$ and $B_T\xrightarrow{p}b$ for a constant $b$, then $A_T+B_T\Rightarrow A+b$. Apply this with \begin{align*} A_T=\tau_T^{\mathrm{DF}}(D),\qquad A=L_D,\qquad B_T=R_T,\qquad b=0. \end{align*} It follows that \begin{align*} \tau_T^{\mathrm{DF}}(D)+R_T\Rightarrow L_D. \end{align*} Using the identity from the previous step, \begin{align*} \tau_T^{\mathrm{ADF}}(D)\Rightarrow L_D. \end{align*} This is exactly the assertion that the augmented Dickey-Fuller statistic has the same limiting distribution as the Dickey-Fuller statistic for the deterministic specification $D$. [/step]