Heine-Borel for Finite Dimensional Normed Vector Spaces

I would like to show that finite-dimensional normed vector spaces have the Heine-Borel property (any subset is compact if and only if it is closed and bounded). I have decided to take the following approach:

1. Show compact implies closed and bounded (I used Hausdorffness to show closedness, and definitions to show boundedness)
2. Show that for any normed vector space $(X, \vert\vert{\cdot}\vert\vert)$ of dimension $n$, there is a homeomorphism $T: (X, \vert\vert{\cdot}\vert\vert) \to (\mathbb{R}^n, \vert\vert{\cdot}\vert\vert_\infty)$.
3. Show Heine-Borel for $\mathbb{R}^n$ (say, Rudin's proof in his "Principles" textbook, for example).
4. Show the preservation of closedness, boundedness and compactness by the homeomorphism $T$ in Step 2, concluding the proof.

In Step 2, while I am able to show bijectivity and continuity of the inverse of $T$, where I define $T$ canonically as taking $\sum_{i=1}^n a_i x_i$ to $\sum_{i=1}^n a_i e_i$, where $\{x_1, \dots, x_n\}$ is a chosen (and fixed) basis of $X$ and $\{e_i | i \in \{1, \dots, n\}\}$ is the standard basis for $\mathbb{R}^n$, I am unable to directly show continuity of $T$.

Note that I do not want to use more advanced results, unless it doesn't assume Bolzano-Weierstrass or Heine-Borel. For instance, I do not want to show that finite-dimensional normed vector spaces are Banach spaces, and using the linearity of $T$ and the continuity of $T^{-1}$ to conclude that $T$ is a homeomorphism by the Open Mapping Theorem, unless all of the intermediate steps can be easily achieved without compactness results or further related work.

The best would be a direct $\varepsilon-\delta$ proof of continuity of $T$ which I don't seem able to come up with.