Is this correct:

Throughout we write
\|x\|:=\sqrt{\Re(x)^2+\Im(x)^2},\qquad
B_n:=\{x\in V_n:\|x\|\le1\}.
A subset A\subset B_n is \varepsilon-dense if every point of B_n lies within Euclidean distance \varepsilon of A.

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Theorem

Let
	•	V_n be complex vector spaces with \dim_{\Bbb C}V_n=m_n\to\infty;
	•	X_n\subset V_n closed algebraic subvarieties with
at most N irreducible components, each parametrised by polynomials of total degree \le d;
	•	X_n \varepsilon-dense in the unit ball B_n\subset V_n for a fixed 0<\varepsilon<1.

Put k_n:=\dim_{\Bbb C}X_n.  Then

\boxed{\displaystyle\lim_{n\to\infty}\frac{k_n}{m_n}=1 }.

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1 Real dimensions and covering numbers

Denote real dimensions

M_n:=2m_n,\quad K_n:=2k_n,\quad c_n’:=M_n-K_n .

For A\subset\Bbb R^{M_n} and \delta>0 let \mathcal N(A,\delta) be the minimum number of closed \delta-balls that cover A.

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2 Metric-entropy bounds
	1.	Density ⇒ tube containment
B_n\subset T_n(\varepsilon):=\{x:d(x,X_n)\le\varepsilon\}, hence
\mathcal N(B_n,\varepsilon)\;\le\;\mathcal N(T_n(\varepsilon),\varepsilon).
\tag{1}
	2.	Packing lower bound (standard volume argument)
\mathcal N(B_n,\varepsilon)\;\ge\;\bigl(C_0/\varepsilon\bigr)^{M_n},
\quad C_0>0\text{ universal}.
\tag{2}
	3.	Yomdin-Gromov upper bound
For every fixed \eta\in(0,C_0] there exists
C_1(d,\eta)\ge1, independent of n, such that each irreducible component Y_i\subset X_n satisfies
\mathcal N\!\bigl(Y_i,{\varepsilon}/{2}\bigr)
\;\le\;
C_1(d,\eta)\Bigl(\tfrac{C_0+\eta}{\varepsilon/2}\Bigr)^{K_n}.
Because X_n has at most N components and T_n(\varepsilon)\subset X_n\oplus B(0,\varepsilon),
\mathcal N(T_n(\varepsilon),\varepsilon)
\;\le\;
N\,C_1(d,\eta)\Bigl(\tfrac{2(C_0+\eta)}{\varepsilon}\Bigr)^{K_n}.
\tag{3}

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3 Global entropy inequality

Combining (1)–(3):

\bigl(C_0/\varepsilon\bigr)^{M_n}
\;\le\;
N\,C_1(d,\eta)
\Bigl(\tfrac{2(C_0+\eta)}{\varepsilon}\Bigr)^{K_n},
\qquad\forall n.
\tag{4}

Take natural logarithms, isolate c_n’:

c_n’\,\log\!\Bigl(\tfrac{C_0}{\varepsilon}\Bigr)
\;\le\;
\log N+\log C_1(d,\eta)
+K_n\log\!\Bigl(\tfrac{2(C_0+\eta)}{C_0}\Bigr).
\tag{5}

The third term on the right is non-negative, so dropping it weakens the inequality:

\boxed{%
c_n’\;\le\;
C_{\max}(N,d,\varepsilon;\eta):=
\frac{\log N+\log C_1(d,\eta)}
{\log(C_0/\varepsilon)}}.
\tag{6}

Key point: C_{\max} is a finite constant independent of n.

Quantitative side note.
Modern estimates give C_1(d,\eta)\le \alpha_1d^{\alpha_2}\eta^{-\beta(d)} with \beta(d)=O(d^\gamma), so
C_{\max}=O\!\bigl(\log N+\log d+d^\gamma\log(1/\eta)\bigr).

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4 Vanishing codimension ratio

Because M_n=2m_n\to\infty while inequality (6) bounds c_n’ by a fixed constant,

\frac{c_n’}{M_n}\;\xrightarrow[n\to\infty]{}\;0.

Consequently

\frac{K_n}{M_n}=1-\frac{c_n’}{M_n}\;\longrightarrow\;1,
\qquad
\frac{k_n}{m_n}=\frac{K_n}{M_n}\;\longrightarrow\;1.

No additional tuning of \eta or contradiction argument is necessary—the bounded-codimension estimate alone forces the limit.

\boxed{\displaystyle\lim_{n\to\infty}\frac{k_n}{m_n}=1}.
\qquad\blacksquare

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5 Immediate corollary

Since the limit equals 1, choose any 0<C<1 (e.g. C=\tfrac12); eventually k_n/m_n>C, so

\limsup_{n\to\infty}\frac{k_n}{m_n}\;\ge\;C.
\qquad\blacksquare

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6 Conceptual recap
	•	The unit ball B_n has metric entropy \asymp (C_0/\varepsilon)^{M_n}, exploding exponentially in every ambient real dimension.
	•	A union of N degree-d algebraic pieces can contribute at most a fixed factor N\,C_1(d,\eta) to its own entropy.
That buys only O(1) real dimensions of slack c_n’.
	•	As M_n grows without bound, a bounded slack means the intrinsic dimension k_n is forced to track the ambient dimension m_n ever more closely, whence k_n/m_n\to1.