Is the following proof valid?
CLAIM:
The space obtained by attaching a disc to a Mobius Strip along the boundary is homeomorphic to the projective plane.
PROOF:
We begin by showing that the boundary of the Mobius Strip $M$ is homeomorphic to $\mathbb{S}^1$. By using the polygon representation of $M$, it is obvious that the manifold boundary is the space $\partial M=([0,1]\times\{0,1\})/\sim$ under the equivalence relation defined by $(1,1)\sim(0,0)$ and $(1,0)\sim(0,1)$. Because $[0,1]\times\{0,1\}$ is compact, $\partial M$ must also be compact. We know from properties of manifold boundaries that this is a 1-manifold, so by the classification theorem, $\partial M\approx\mathbb{S}^1$. Thus, it is possible to attach $\partial\overline{\mathbb{B}}^2$ to $\partial M$ through an adjunction space.
We now show that this space is homeomorphic to $\mathbb{P}^2$ using polygon representations. We represent the two spaces as follows:
$$\overline{\mathbb{B}}^2=\langle b,c|bcb^{-1}\rangle=\langle a,b,c|bca,a^{-1}cb{-1}\rangle$$
$$M=\langle a,b',c'|ab'ac'\rangle$$
Next, we align the $c$ edge of $bca$ with the $b'$ edge of $M$, and the $c$ edge of $a^{-1}cb^{-1}$ with the $c'$ edge of $M$. We do this in such a way that the $c$ edges of our representation of the disc face the same direction. Note that this is equivalent to attaching $\overline{\mathbb{B}}^2$ along the edge of $M$. Then, our polygon representation of the resulting shape is as follows:
$$\langle a,b|baabaa\rangle$$
Using elementary transformations:
$$\langle a,b|baabaa\rangle=\langle a,b,c|baac,c^{-1}baa\rangle=\langle a,b,c|cbaa,a^{-1}a^{-1}b^{-1}c\rangle=\langle a,b,c|cbaa^{-1}b^{-1}c\rangle=\langle c|cc\rangle$$
But $\langle c|cc\rangle$ is homeomorphic to $\mathbb{P}^2$, thus, $M\cup_f \overline{\mathbb{B}}^2\approx \mathbb{P}^2$, where $f$ attaches the two spaces along the boundary.
$\blacksquare$
I'm pretty sure the proof is correct. My main concern is when the disc is attached to the Mobius Strip. I believe I did it in a way that keeps the disc's boundary going the right direction, but I'm unsure if my reasoning for it is rigorous enough.