I have shown that the correspondence $f: X \rightarrow \mathbb{R} \oplus X$ defines an embedding of the Grassmannian $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$ into $\mathrm{Gr}_{k+1}(\mathbb{R} \oplus \mathbb{R}^{n+k})=\mathrm{Gr}_{k+1}(\mathbb{R}^{n+k+1})$. Indeed, this identifies an isomorphism that carries an $r$-cell of $\mathrm{Gr}_{n}(\mathbb{R}^{m})$ onto the $r$-cell of $\mathrm{Gr}_{n+1}(\mathbb{R}^{m+1})$ corresponding to the same partition. (This is Problem 6-C in Milnor- Stasheff).
From this, I wish to prove that $f$ takes the $\left | \underline{a} \right |$-cell of $\mathrm{Gr}_{k}(\mathbb{R}^{n+k})$, which corresponds to the given Schubert symbol $\underline{a}$, onto the $\left | \underline{a} \right |$-cell of $\mathrm{Gr}_{k+1}(\mathbb{R}^{n+k+1})$, which corresponds to the same Schubert symbol $\underline{a}$. How might this be done? Any help would be appreciated.
First, for clarity, write $f(X) = \mathbf{R}\oplus X$. To talk of Schubert cells you must first fix an ordered basis (or a full flag, equivalently). Next you must think of a compatible full flags in $\mathbf{R}^n$ and $\mathbf{R}^{n+1}$, (this is implicit in your definition of $f$) then the correspondence in Schubert cells will follow.
(Added for clarification)
If $v_1,v_2,\ldots,v_{n+k}$ is the ordered basis of $\mathbf{R}^{n+k}$, take $v,v_1,v_2,\ldots,v_{n+k}$ in that order as basis of $\mathbf{R}^{n+k+1}$. So an element of Gr$_k(\mathbf{R}^{n+k})$ along with $v$ generates a $k+1$-dim'l subspace, and that is your embedding. The tuple $a=(a_1,a_2,\ldots, a_k)$ specifies where the dimension jumps occur for a k-dim'l space intersected with terms of full flag from the ordered basis (giving a Schubert cell). For a non-decreasing seq. of numbers with jumps at specified slots adding a constant to each term means jumps still occur at the same slots.