[proofplan]
Choose a smooth connection on the real bundle $E$ and extend it complex-linearly to a connection on $E_{\mathbb C}$. The de Rham Chern class $c_{2k}(E_{\mathbb C})$ is represented by the $2k$-th Chern-Weil form of this complexified connection. By the defining Chern-Weil convention for Pontryagin classes, the $k$-th Pontryagin form of $E$ is $(-1)^k$ times that same $2k$-th Chern form. Passing to de Rham cohomology gives the claimed identity.
[/proofplan]
[step:Choose a real connection and complexify it]
Because $M$ is paracompact, the smooth real vector bundle $E\to M$ admits a smooth real connection. Fix one such connection and denote it by $\nabla^E$. Define the complexified bundle
\begin{align*}
E_{\mathbb C}:=E\otimes_{\mathbb R}\mathbb C.
\end{align*}
The connection $\nabla^E$ extends uniquely by complex linearity to a smooth complex-linear connection
\begin{align*}
\nabla^{\mathbb C}:\Gamma(E_{\mathbb C})\to \Omega^1(M;E_{\mathbb C}).
\end{align*}
Let
\begin{align*}
F_{\nabla^{\mathbb C}}\in \Omega^2(M;\operatorname{End}_{\mathbb C}(E_{\mathbb C}))
\end{align*}
denote the curvature of this complex connection.
[/step]
[step:Represent the even Chern class by its Chern-Weil form]
For each integer $j\ge 0$, let
\begin{align*}
c_j(E_{\mathbb C},\nabla^{\mathbb C})\in \Omega^{2j}(M;\mathbb C)
\end{align*}
denote the $j$-th Chern-Weil Chern form of the connection $\nabla^{\mathbb C}$. By the [Chern-Weil construction of Chern classes](/theorems/9769), as in [citetheorem:9769], the form $c_j(E_{\mathbb C},\nabla^{\mathbb C})$ is closed and represents the de Rham Chern class:
\begin{align*}
c_j(E_{\mathbb C})=[c_j(E_{\mathbb C},\nabla^{\mathbb C})]_{\mathrm{dR}}.
\end{align*}
Applying this with $j=2k$ gives
\begin{align*}
c_{2k}(E_{\mathbb C})=[c_{2k}(E_{\mathbb C},\nabla^{\mathbb C})]_{\mathrm{dR}}.
\end{align*}
[guided]
The point of introducing the connection is that de Rham Chern classes are represented by differential forms produced from curvature. For the complex vector bundle $E_{\mathbb C}\to M$ and the complex-linear connection $\nabla^{\mathbb C}$, the Chern-Weil construction assigns to every integer $j\ge 0$ a closed form
\begin{align*}
c_j(E_{\mathbb C},\nabla^{\mathbb C})\in \Omega^{2j}(M;\mathbb C).
\end{align*}
The theorem [citetheorem:9769] applies because $E_{\mathbb C}$ is a smooth complex vector bundle of finite rank and $\nabla^{\mathbb C}$ is a smooth complex-linear connection. Its conclusion is precisely that this closed Chern form represents the de Rham Chern class of the bundle:
\begin{align*}
c_j(E_{\mathbb C})=[c_j(E_{\mathbb C},\nabla^{\mathbb C})]_{\mathrm{dR}}.
\end{align*}
We need the even index $j=2k$, because the $k$-th Pontryagin class has degree $4k$, while the $j$-th Chern class has degree $2j$. Substituting $j=2k$ therefore gives
\begin{align*}
c_{2k}(E_{\mathbb C})=[c_{2k}(E_{\mathbb C},\nabla^{\mathbb C})]_{\mathrm{dR}}.
\end{align*}
[/guided]
[/step]
[step:Compare the Pontryagin form convention with the even Chern form]
By the standard de Rham Chern-Weil convention for Pontryagin classes of a real vector bundle, the $k$-th Pontryagin form of $E$ associated to the real connection $\nabla^E$ is defined by
\begin{align*}
p_k(E,\nabla^E):=(-1)^k c_{2k}(E_{\mathbb C},\nabla^{\mathbb C})\in \Omega^{4k}(M;\mathbb C).
\end{align*}
The connection-independence of Pontryagin forms, as in [citetheorem:9777], identifies the de Rham Pontryagin class with the cohomology class of this form:
\begin{align*}
p_k(E)=[p_k(E,\nabla^E)]_{\mathrm{dR}}.
\end{align*}
Substituting the defining formula for $p_k(E,\nabla^E)$ gives
\begin{align*}
p_k(E)=(-1)^k[c_{2k}(E_{\mathbb C},\nabla^{\mathbb C})]_{\mathrm{dR}}.
\end{align*}
[/step]
[step:Identify the resulting class with $(-1)^k c_{2k}(E_{\mathbb C})$]
From the Chern-Weil representative identity for $c_{2k}(E_{\mathbb C})$,
\begin{align*}
[c_{2k}(E_{\mathbb C},\nabla^{\mathbb C})]_{\mathrm{dR}}=c_{2k}(E_{\mathbb C}).
\end{align*}
Therefore
\begin{align*}
p_k(E)=(-1)^k c_{2k}(E_{\mathbb C})
\end{align*}
in $H_{\mathrm{dR}}^{4k}(M;\mathbb C)$. This is the desired formula for every integer $k\ge 0$.
[/step]
