I have an iterative method \begin{eqnarray} X_{k+1}=(1+\beta)X_k-\beta X_k A X_k~~~~~~~~~~~~~~~~~ k = 0,1,\ldots \end{eqnarray} with initial approximation $X_0 = \beta A^*$ ($\beta$ is scalar lying between 0 to 1)converging to the moore-penrose inverse $A^+$. $\{X_{k}\}$ are sequence of approximations. I have to prove that $\{X_{k}\}$ defined by above equations satisfies $R(X_k)= R(A^*)$ and $N(X_k)=N(A^*)$ for $k\geq 0$. Where $R(X_k)$, $N(X_k)$ denotes the range and null space of $\{X_{k}\}$ .I am using mathematical induction to prove above result.This is what i did.
It trivially holds for $k = 0$ (since for $k = 0$,$X_0 = \beta A^*$). Suppose result is true for some k. Let $y\in N(X _k)$ be an arbitrary vector. From above equation we have \begin{eqnarray*} X_{k+1} y = (1+\beta)X_{k}y - \beta X_{k}AX_{k}y = 0 \end{eqnarray*} (since $X_{k}y = 0$). This gives $y\in ~N(X_{k+1})$ thus $N(X_k)\subseteq N(X_{k+1})$ holds true. Now my question is how the statement $R(X_k)\supseteq R(X_{k+1})$ can be proved analogously. I tried much but unable to figure out how to prove $R(X_k)\supseteq R(X_{k+1})$. Little hint will work for me. I would be very much thankful.