Define $H^+=\lim_{\delta \to 0^+} (\delta I+H^*H)^{-1}H^*$ as Moore-Penrose Pseudoinverse of arbitrary matrix $H$. How can we show that $HH^+H=H$.i.e. it satisfies the first property of Moore-Penrose Pseudoinverse.
I have tried the following: $$ \begin{align} HH^+H= & H (\lim_{\delta \to 0^+} (\delta I+H^*H)^{-1}H^*)H\\ =& H (\lim_{\delta \to 0^+} (\delta I+H^*H)^{-1}H^*H)\\ =& H (\lim_{\delta \to 0^+} (\delta I+H^*H)^{-1})(\lim_{\delta \to 0^+} (\delta I+H^*H)))\\ =& H \lim_{\delta \to 0^+} \bigg[(\delta I+H^*H)^{-1} (\delta I+H^*H)\bigg]\\ =&H. \end{align}$$
Is it alright? Anyone hints or suggestions?