Proof that the Blank Tape Halting Problem is undecideable
You need to explain how to construct $R$. In your case this can be done formally, by constructing a machine which works as follows:


Write $w$ on the input tape.
Move the head back to its initial position.
Transfer control to $T$.


It is not too hard to construct $R$ explicitly from $T$.

In order for your argument to go through, you need to show that this reduction is computable. Usually we describe such reductions informally and just claim (without proof) that it is computable, since actually proving such things formally is extremely technical and longwinded.