[proofplan]
Fix an arbitrary ordinal $\lambda$ and choose a cardinal $\theta$ large enough that every set in $V_\lambda$ has a [transitive closure](/theorems/1493) of size at most $\theta$, with $\theta>\lambda$. Supercompactness gives an elementary embedding $j:V\to M$ with critical point $\kappa$, $j(\kappa)>\theta$, and $M^\theta\subset M$. We then show that every $x\in V_\lambda$ belongs to $M$ by coding the pointed membership structure on $\operatorname{TC}(\{x\})$ as a $\theta$-sequence of ordinals. Since this code lies in $M$ and the Mostowski collapse of well-founded extensional structures is absolute between transitive classes, $M$ decodes the distinguished point as $x$ itself.
[/proofplan]
[step:Choose a supercompactness embedding above the required rank]
Let $\lambda$ be an arbitrary ordinal. Choose an ordinal $\lambda_0$ such that $\lambda\leq \lambda_0$ and $\kappa\leq \lambda_0$. It is enough to prove $V_{\lambda_0}\subset M$, since then $V_\lambda\subset V_{\lambda_0}\subset M$.
Choose a cardinal $\theta$ such that $\theta>\lambda_0$ and $|V_{\lambda_0}|\leq \theta$. Since $\kappa$ is supercompact, there exist a transitive class $M$ and an elementary embedding
\begin{align*}
j: V \to M
\end{align*}
such that $\operatorname{crit}(j)=\kappa$, $j(\kappa)>\theta$, and $M^\theta\subset M$.
Because $\theta>\lambda_0\geq\lambda$, the same embedding will witness $\lambda$-strongness once we prove $V_{\lambda_0}\subset M$.
[/step]
[step:Code the pointed transitive closure of a set in $V_{\lambda_0}$]
Fix $x\in V_{\lambda_0}$. Define
\begin{align*}
T_x:=\operatorname{TC}(\{x\}).
\end{align*}
Since $x\in V_{\lambda_0}$, every element of $T_x$ belongs to $V_{\lambda_0}$, so $|T_x|\leq |V_{\lambda_0}|\leq\theta$.
Choose an ordinal $\mu_x\leq\theta$ and a bijection
\begin{align*}
b_x:\mu_x\to T_x.
\end{align*}
Let $p_x<\mu_x$ be the unique ordinal such that $b_x(p_x)=x$. Define the binary relation
\begin{align*}
E_x\subset \mu_x\times\mu_x
\end{align*}
by
\begin{align*}
\alpha\,E_x\,\beta \iff b_x(\alpha)\in b_x(\beta).
\end{align*}
The structure $(\mu_x,E_x,p_x)$ is well-founded and extensional, because it is isomorphic via $b_x$ to the pointed membership structure $(T_x,\in,x)$.
[guided]
We want to recover $x$ inside $M$, but we cannot assume $x\in M$ at this point. The way around this is to replace $x$ by an ordinal code for the entire membership structure generated by $x$.
Define
\begin{align*}
T_x:=\operatorname{TC}(\{x\}).
\end{align*}
This is the smallest transitive set containing $x$. Since $x\in V_{\lambda_0}$, all members that appear by iterating membership downward from $x$ still lie in $V_{\lambda_0}$. Therefore $T_x\subset V_{\lambda_0}$, and hence
\begin{align*}
|T_x|\leq |V_{\lambda_0}|\leq\theta.
\end{align*}
Choose an ordinal $\mu_x\leq\theta$ and a bijection
\begin{align*}
b_x:\mu_x\to T_x.
\end{align*}
The point corresponding to $x$ is the unique ordinal $p_x<\mu_x$ with $b_x(p_x)=x$. We now transfer the membership relation on $T_x$ back to $\mu_x$ by defining
\begin{align*}
E_x\subset \mu_x\times\mu_x
\end{align*}
through
\begin{align*}
\alpha\,E_x\,\beta \iff b_x(\alpha)\in b_x(\beta).
\end{align*}
The map $b_x$ is an isomorphism from $(\mu_x,E_x)$ onto $(T_x,\in)$. Since ordinary membership on a transitive set is well-founded and extensional, the coded relation $E_x$ is also well-founded and extensional. The distinguished point $p_x$ records exactly where $x$ sits in this coded structure.
[/guided]
[/step]
[step:Place the ordinal code inside the target model]
Encode the triple $(\mu_x,E_x,p_x)$ by a function
\begin{align*}
c_x:\theta\to \theta\cup\{\theta\}
\end{align*}
using any fixed definable pairing function on $\theta$ and a characteristic function for $E_x$, together with the parameters $\mu_x$ and $p_x$. Thus $c_x$ is a $\theta$-sequence of ordinals belonging to $M$: every ordinal below or equal to $\theta$ lies in $M$, since $j(\kappa)>\theta$ and $j$ fixes all ordinals below $\kappa$ while $M$ is transitive and contains the ordinal $\theta$.
By the closure assumption $M^\theta\subset M$, the sequence $c_x$ belongs to $M$. Since the decoding procedure from $c_x$ to $(\mu_x,E_x,p_x)$ is first-order definable, $M$ contains the coded structure $(\mu_x,E_x,p_x)$.
[/step]
[step:Decode the code by the absolute Mostowski collapse]
In $V$, the Mostowski collapse of the well-founded extensional structure $(\mu_x,E_x)$ is exactly $(T_x,\in)$, with collapsing map
\begin{align*}
\pi_x:\mu_x\to T_x
\end{align*}
equal to $b_x$. Hence $\pi_x(p_x)=x$.
Since $M$ is transitive and contains $(\mu_x,E_x,p_x)$, well-foundedness and extensionality of $E_x$ are absolute between $M$ and $V$. Therefore the Mostowski collapse computed inside $M$ is the same collapse computed in $V$. Consequently $M$ contains the collapsed value of the distinguished point $p_x$, namely
\begin{align*}
\pi_x(p_x)=x.
\end{align*}
Thus $x\in M$.
[/step]
[step:Conclude that the supercompactness embedding witnesses strongness]
The element $x\in V_{\lambda_0}$ was arbitrary, so $V_{\lambda_0}\subset M$. Since $V_\lambda\subset V_{\lambda_0}$, we also have $V_\lambda\subset M$. The embedding $j:V\to M$ satisfies $\operatorname{crit}(j)=\kappa$ and
\begin{align*}
j(\kappa)>\theta>\lambda.
\end{align*}
Therefore, for the arbitrary ordinal $\lambda$, there is an elementary embedding $j:V\to M$ witnessing $\lambda$-strongness of $\kappa$. Hence every supercompact cardinal is strong.
[/step]
