[proofplan] By the [Finite Type of Compact Manifolds](/theorems/1532), $M$ admits a finite good cover — a finite open cover $\{V_1, \dots, V_n\}$ whose every non-empty finite intersection is diffeomorphic to $\mathbb{R}^d$. We prove finite-dimensionality of de Rham cohomology by induction on the cardinality $n$ of such a good cover, treating any smooth manifold equipped with a good cover of size $n$. The base case $n = 1$ reduces $M$ to $\mathbb{R}^d$, where the Poincaré lemma computes the cohomology explicitly. The inductive step splits the cover into a single piece $U$ and the union $W$ of the remaining pieces; the intersection $U \cap W$ inherits a strictly smaller good cover, so the inductive hypothesis applies to $W$ and to $U \cap W$. The [Mayer-Vietoris](/theorems/1533) short exact sequence of de Rham complexes then yields, via the [Long Exact Cohomology Sequence](/theorems/3471), an exact triangle in which $H^k_{\mathrm{dR}}(M)$ is sandwiched between finite-dimensional spaces, forcing it to be finite-dimensional itself. [/proofplan]

[step:Reduce to the existence of a finite good cover on $M$]Since $M$ is smooth, compact, and without boundary, $M$ is a closed manifold. Applying the [Finite Type of Compact Manifolds](/theorems/1532), there exists a *finite good cover* of $M$: a finite collection \begin{align*} \mathcal{V} = \{V_1, V_2, \dots, V_n\} \end{align*} of open subsets $V_i \subseteq M$ such that $M = \bigcup_{i=1}^n V_i$ and for every non-empty subset $S \subseteq \{1, \dots, n\}$ the intersection \begin{align*} V_S := \bigcap_{i \in S} V_i \end{align*} is either empty or diffeomorphic (as a smooth manifold) to $\mathbb{R}^d$.[/step]

[guided]The first reduction is to replace the abstract compactness hypothesis by a concrete combinatorial structure on $M$: a finite good cover. The key payoff is twofold. First, the cover is *finite*, so we can do induction on its size. Second, every non-empty finite intersection of the cover is *diffeomorphic to $\mathbb{R}^d$*; this is the crucial input for the base case, where we will compute the cohomology of a piece of the cover directly. The [Finite Type of Compact Manifolds](/theorems/1532) requires that $M$ be either a closed manifold (compact, without boundary) or the interior of a compact manifold with boundary. Our hypothesis — compact and without boundary — places $M$ in the first class. The construction inside that theorem uses geodesic balls of a Riemannian metric (which exists by a standard [partition of unity](/page/Partition%20of%20Unity) argument) and exploits the fact that small geodesic balls are geodesically convex, hence diffeomorphic to $\mathbb{R}^d$, and their finite intersections are again geodesically convex. So we have a finite good cover $\mathcal{V} = \{V_1, \dots, V_n\}$ of $M$, with the property that for every non-empty index set $S \subseteq \{1, \dots, n\}$, the intersection $V_S = \bigcap_{i \in S} V_i$ is either empty or diffeomorphic to $\mathbb{R}^d$.[/guided]

[step:Set up the induction on the cardinality of a good cover] For each integer $n \ge 1$ define the statement $P(n)$: *for every smooth manifold $N$ admitting a finite good cover of cardinality at most $n$, the de Rham cohomology $H^k_{\mathrm{dR}}(N)$ is a finite-dimensional real [vector space](/page/Vector%20Space) for every $k \ge 0$.* Here a *good cover of cardinality at most $n$* on $N$ means a finite open cover $\{W_1, \dots, W_m\}$ of $N$ with $m \le n$ and every non-empty intersection $W_{i_1} \cap \dots \cap W_{i_r}$ diffeomorphic to $\mathbb{R}^{\dim N}$. We will prove $P(n)$ for all $n \ge 1$ by induction. By Step 1 applied to $M$, the statement $P(n)$ for the cardinality of the good cover $\mathcal V$ produced there implies $H^k_{\mathrm{dR}}(M) < \infty$ for every $k$, which is the conclusion of the theorem. [/step]

[step:Establish the base case using the Poincaré Lemma for $\mathbb{R}^d$]We verify $P(1)$. Let $N$ be a smooth manifold admitting a good cover of cardinality $1$. Then $N = W_1$ where $W_1$ is diffeomorphic to $\mathbb{R}^{\dim N}$. Since de Rham cohomology is a diffeomorphism invariant, it suffices to compute $H^k_{\mathrm{dR}}(\mathbb{R}^{\dim N})$. By the Poincaré lemma for $\mathbb{R}^d$ (citing a result not yet in the wiki: *Poincaré Lemma — every closed form on a star-shaped open subset of $\mathbb{R}^d$ is exact*, applied to the convex set $\mathbb{R}^d$), \begin{align*} H^k_{\mathrm{dR}}(\mathbb{R}^{\dim N}) = \begin{cases} \mathbb{R}, & k = 0, \\ 0, & k \ge 1, \end{cases} \end{align*} where the $k = 0$ case follows because the kernel of $d : \Omega^0(\mathbb{R}^{\dim N}) \to \Omega^1(\mathbb{R}^{\dim N})$ is the space of locally constant functions on the connected manifold $\mathbb{R}^{\dim N}$, which is $\mathbb{R} \cdot 1$. In every degree $H^k_{\mathrm{dR}}(N)$ is finite-dimensional. Thus $P(1)$ holds.[/step]

[guided]The base case is where the structure of a good cover becomes geometric content. The single [open set](/page/Open%20Set) $W_1$ is diffeomorphic to $\mathbb{R}^{\dim N}$ — not merely homotopy equivalent — and we can pull back the de Rham complex via the diffeomorphism. This is legitimate because the [exterior derivative](/theorems/1525) commutes with pullback, so a diffeomorphism $\Phi: N \to \mathbb{R}^d$ induces an isomorphism of cochain complexes $\Phi^*: \Omega^\bullet(\mathbb{R}^d) \xrightarrow{\sim} \Omega^\bullet(N)$ and hence isomorphisms on cohomology $H^k_{\mathrm{dR}}(\mathbb{R}^d) \cong H^k_{\mathrm{dR}}(N)$. The Poincaré lemma is the statement that on a star-shaped open subset $U \subseteq \mathbb{R}^d$ — in particular on $\mathbb{R}^d$ itself — every closed differential form of positive degree is exact. The standard proof builds a homotopy operator $K: \Omega^k(U) \to \Omega^{k-1}(U)$ via radial integration and verifies the chain homotopy identity $dK + Kd = \mathrm{id}$ on $\Omega^k(U)$ for $k \ge 1$. From this one reads off $H^k_{\mathrm{dR}}(\mathbb{R}^d) = 0$ for $k \ge 1$. For $k = 0$: closed $0$-forms are smooth functions $f$ with $df = 0$, i.e. $\partial_{x_i} f = 0$ for every $i$, which on the connected manifold $\mathbb{R}^d$ forces $f$ to be a constant. The space of constants is the $1$-dimensional [vector space](/page/Vector%20Space) $\mathbb{R} \cdot 1$. Since there are no $(-1)$-forms, no quotient is taken, and $H^0_{\mathrm{dR}}(\mathbb{R}^d) = \mathbb{R}$. Either way the dimension is at most $1$, so $H^k_{\mathrm{dR}}(N)$ is finite-dimensional in every degree.[/guided]

[step:Split the good cover and inherit smaller good covers on $U$, $W$, and $U \cap W$]Assume $P(n-1)$ for some $n \ge 2$, and let $N$ be a smooth manifold equipped with a good cover $\{W_1, \dots, W_n\}$ of cardinality $n$. Define the open subsets \begin{align*} U &:= W_n \subseteq N, & W &:= \bigcup_{i=1}^{n-1} W_i \subseteq N. \end{align*} Both are open submanifolds of $N$, and $N = U \cup W$. We exhibit good covers of $W$, $U$, and $U \cap W$: **Good cover of $W$.** The collection $\{W_1, \dots, W_{n-1}\}$ is by construction an open cover of $W$, with cardinality $n-1$. For any non-empty $S \subseteq \{1, \dots, n-1\}$, the intersection $\bigcap_{i \in S} W_i$ is either empty or diffeomorphic to $\mathbb{R}^{\dim N}$ because $\{W_1, \dots, W_n\}$ is a good cover of $N$. Hence $\{W_1, \dots, W_{n-1}\}$ is a good cover of $W$ of cardinality $n - 1$. **Good cover of $U$.** The collection $\{U\}$ is a good cover of $U$ of cardinality $1$, since $U = W_n$ is diffeomorphic to $\mathbb{R}^{\dim N}$ (this is the $S = \{n\}$ case of the good cover property of $\mathcal V$ on $N$). **Good cover of $U \cap W$.** We have \begin{align*} U \cap W = W_n \cap \bigcup_{i=1}^{n-1} W_i = \bigcup_{i=1}^{n-1} \bigl(W_i \cap W_n\bigr), \end{align*} so the collection $\{W_1 \cap W_n, \dots, W_{n-1} \cap W_n\}$ is an open cover of $U \cap W$ of cardinality at most $n - 1$ (discarding empty members preserves the cover and only decreases the cardinality). For any non-empty $S \subseteq \{1, \dots, n-1\}$, \begin{align*} \bigcap_{i \in S} (W_i \cap W_n) = \Bigl(\bigcap_{i \in S} W_i\Bigr) \cap W_n = \bigcap_{i \in S \cup \{n\}} W_i, \end{align*} which is either empty or diffeomorphic to $\mathbb{R}^{\dim N}$ since $\mathcal V$ is a good cover of $N$. Thus $\{W_i \cap W_n\}_{i=1}^{n-1}$, after discarding empty entries, is a good cover of $U \cap W$ of cardinality $\le n - 1$. By the inductive hypothesis $P(n - 1)$ applied to $W$ and to $U \cap W$, and by the base case $P(1)$ applied to $U$: \begin{align*} \dim_{\mathbb{R}} H^k_{\mathrm{dR}}(U), \quad \dim_{\mathbb{R}} H^k_{\mathrm{dR}}(W), \quad \dim_{\mathbb{R}} H^k_{\mathrm{dR}}(U \cap W) < \infty \qquad \text{for every } k \ge 0. \end{align*}[/step]

[guided]This step is the bookkeeping engine of the induction. We need to split $N$ as the union of two open sets $U \cup W$, and we need *both* pieces and their intersection to be controlled by the inductive hypothesis. The trick is to peel off one element of the cover at a time: take $U = W_n$, the last set, and $W$ the union of the preceding $n - 1$ sets. Why does each piece have a strictly smaller good cover? Three verifications, one per piece: (i) For $W = W_1 \cup \dots \cup W_{n-1}$: the original sets $W_1, \dots, W_{n-1}$ themselves form an open cover of $W$. Their intersections are intersections of the original good cover of $N$, hence empty or diffeomorphic to $\mathbb{R}^{\dim N}$. So $\{W_1, \dots, W_{n-1}\}$ is a good cover of $W$ of cardinality exactly $n - 1 < n$. (ii) For $U = W_n$: a single set, diffeomorphic to $\mathbb{R}^{\dim N}$. This is the $S = \{n\}$ case of the good cover property and gives a good cover $\{U\}$ of cardinality $1$. (iii) For $U \cap W$: distribute the intersection over the union, \begin{align*} U \cap W = W_n \cap \bigcup_{i=1}^{n-1} W_i = \bigcup_{i=1}^{n-1} (W_i \cap W_n). \end{align*} The candidate cover is $\{W_i \cap W_n\}_{i=1}^{n-1}$. The crucial property we need is that intersections among elements of this candidate cover are again controlled. Computing, \begin{align*} \bigcap_{i \in S} (W_i \cap W_n) = \bigcap_{i \in S \cup \{n\}} W_i, \end{align*} which is empty or diffeomorphic to $\mathbb{R}^{\dim N}$ because $\mathcal V$ is a good cover of $N$. So $\{W_i \cap W_n\}_{i=1}^{n-1}$ is a good cover of $U \cap W$, and its cardinality is at most $n - 1$. Note that the cardinality could strictly decrease if some $W_i \cap W_n$ are empty (we discard them), but we only need the bound $\le n - 1$ to invoke $P(n - 1)$. The inductive hypothesis $P(n - 1)$ now applies to both $W$ and $U \cap W$, and the base case $P(1)$ applies to $U$. We conclude all three de Rham cohomology spaces are finite-dimensional in every degree.[/guided]

[step:Apply the Mayer–Vietoris long exact sequence and bound the dimension of $H^k_{\mathrm{dR}}(N)$]We now combine the finite-dimensionality of $H^\bullet_{\mathrm{dR}}(U)$, $H^\bullet_{\mathrm{dR}}(W)$, and $H^\bullet_{\mathrm{dR}}(U \cap W)$ from Step 4 with the Mayer–Vietoris machinery. The pair $(U, W)$ consists of open subsets of the smooth manifold $N$ with $N = U \cup W$. The hypotheses of [Mayer-Vietoris](/theorems/1533) are therefore satisfied, and we obtain a short exact sequence of cochain complexes \begin{align*} 0 \longrightarrow \Omega^\bullet(N) \xrightarrow{\ \alpha\ } \Omega^\bullet(U) \oplus \Omega^\bullet(W) \xrightarrow{\ \beta\ } \Omega^\bullet(U \cap W) \longrightarrow 0, \end{align*} where $\alpha(\omega) = (\omega|_U, \omega|_W)$ and $\beta(\tilde\alpha, \tilde\beta) = \tilde\alpha|_{U \cap W} - \tilde\beta|_{U \cap W}$. The hypotheses of the [Long Exact Cohomology Sequence](/theorems/3471) are precisely that one has a short exact sequence of cochain complexes; we have just established this. Applying it produces, for each $k \ge 0$, an exact sequence of real vector spaces \begin{align*} H^{k-1}_{\mathrm{dR}}(U \cap W) \xrightarrow{\ \delta\ } H^k_{\mathrm{dR}}(N) \xrightarrow{\ \alpha^*\ } H^k_{\mathrm{dR}}(U) \oplus H^k_{\mathrm{dR}}(W), \end{align*} where $\delta$ is the connecting homomorphism and $\alpha^*$ is the map induced by $\alpha$ on cohomology, and exactness at the middle term reads $\ker \alpha^* = \operatorname{im}\,\delta$. (For $k = 0$, the leftmost term is $0$ and the same exactness statement holds with the convention $H^{-1}_{\mathrm{dR}} = 0$.) From exactness at $H^k_{\mathrm{dR}}(N)$, \begin{align*} \dim_{\mathbb{R}} H^k_{\mathrm{dR}}(N) &= \dim_{\mathbb{R}} \ker \alpha^* + \dim_{\mathbb{R}} \operatorname{im}\,\alpha^* \\ &= \dim_{\mathbb{R}} \operatorname{im}\,\delta + \dim_{\mathbb{R}} \operatorname{im}\,\alpha^*. \end{align*} Bounding each image by the dimension of its source, \begin{align*} \dim_{\mathbb{R}} \operatorname{im}\,\delta &\le \dim_{\mathbb{R}} H^{k-1}_{\mathrm{dR}}(U \cap W), \\ \dim_{\mathbb{R}} \operatorname{im}\,\alpha^* &\le \dim_{\mathbb{R}}\bigl(H^k_{\mathrm{dR}}(U) \oplus H^k_{\mathrm{dR}}(W)\bigr) = \dim_{\mathbb{R}} H^k_{\mathrm{dR}}(U) + \dim_{\mathbb{R}} H^k_{\mathrm{dR}}(W), \end{align*} we obtain \begin{align*} \dim_{\mathbb{R}} H^k_{\mathrm{dR}}(N) \le \dim_{\mathbb{R}} H^{k-1}_{\mathrm{dR}}(U \cap W) + \dim_{\mathbb{R}} H^k_{\mathrm{dR}}(U) + \dim_{\mathbb{R}} H^k_{\mathrm{dR}}(W). \end{align*} By Step 4 the right-hand side is finite. Hence $\dim_{\mathbb{R}} H^k_{\mathrm{dR}}(N) < \infty$ for every $k \ge 0$, completing the inductive step. By induction $P(n)$ holds for all $n \ge 1$. Returning to the manifold $M$ of the theorem: by Step 1, $M$ admits a finite good cover of some cardinality $n_0$; applying $P(n_0)$ yields $\dim_{\mathbb{R}} H^k_{\mathrm{dR}}(M) < \infty$ for every $k \ge 0$, as claimed.[/step]

[guided]The final step is where the topology of $N$ assembles from the topology of the pieces $U$, $W$, $U \cap W$. The mechanism is [Mayer-Vietoris](/theorems/1533), which on de Rham complexes takes the form of a short exact sequence of cochain complexes induced by restriction and "subtraction-of-restrictions." We need to check the hypotheses of theorem 1533: $N$ a smooth manifold (yes, by construction in the induction), and $N = U \cup W$ with $U, W$ open (yes, by Step 4). Conclusion: a short exact sequence \begin{align*} 0 \to \Omega^\bullet(N) \to \Omega^\bullet(U) \oplus \Omega^\bullet(W) \to \Omega^\bullet(U \cap W) \to 0 \end{align*} of cochain complexes — exactness as cochain complexes is the content of Mayer-Vietoris on de Rham complexes; surjectivity of the rightmost map relies on a [partition of unity](/page/Partition%20of%20Unity) subordinate to $\{U, W\}$. Next we feed this into the [Long Exact Cohomology Sequence](/theorems/3471), whose sole hypothesis is a short exact sequence of cochain complexes. Conclusion: a long exact sequence in cohomology \begin{align*} \cdots \to H^{k-1}_{\mathrm{dR}}(U \cap W) \xrightarrow{\delta} H^k_{\mathrm{dR}}(N) \xrightarrow{\alpha^*} H^k_{\mathrm{dR}}(U) \oplus H^k_{\mathrm{dR}}(W) \xrightarrow{\beta^*} H^k_{\mathrm{dR}}(U \cap W) \to \cdots \end{align*} We only need exactness at the middle term $H^k_{\mathrm{dR}}(N)$. The dimension estimate is the standard "sandwich" argument for exact sequences. Pick an exact triangle of vector spaces $A \to B \to C$ exact at $B$. Then $B$ is filtered: \begin{align*} 0 \subseteq \ker(B \to C) \subseteq B, \end{align*} with $B / \ker(B \to C) \cong \operatorname{im}(B \to C)$ and $\ker(B \to C) = \operatorname{im}(A \to B)$ by exactness. Hence \begin{align*} \dim B = \dim \operatorname{im}(A \to B) + \dim \operatorname{im}(B \to C) \le \dim A + \dim C. \end{align*} Apply this with $A = H^{k-1}_{\mathrm{dR}}(U \cap W)$, $B = H^k_{\mathrm{dR}}(N)$, $C = H^k_{\mathrm{dR}}(U) \oplus H^k_{\mathrm{dR}}(W)$, and use $\dim(C_1 \oplus C_2) = \dim C_1 + \dim C_2$ for the right factor. By Step 4 all three of these are finite-dimensional, so $\dim_{\mathbb{R}} H^k_{\mathrm{dR}}(N) < \infty$. This closes the induction. Concretely, $P(n - 1) \implies P(n)$ for all $n \ge 2$, and $P(1)$ was established in Step 3. By induction $P(n)$ holds for every $n \ge 1$. To conclude the proof of the theorem: Step 1 produced a finite good cover of $M$, of some cardinality $n_0 \ge 1$. The statement $P(n_0)$, now proved, applies to $M$ and yields $\dim_{\mathbb{R}} H^k_{\mathrm{dR}}(M) < \infty$ for every $k \ge 0$. Since $\Omega^k(M) = 0$ for $k > d$, we automatically have $H^k_{\mathrm{dR}}(M) = 0$ in those degrees, so finite-dimensionality is non-trivial only in the finite range $0 \le k \le d$.[/guided]