PomeCorollary 16.9.38.B1559025EC82 Let $\varphi _0,\varphi _1\in \mathrm{PSH}(X,\theta )$. Define $\varphi _ t=t\varphi _1+(1-t)\varphi _0$ for $t\in (0,1)$. Then
\[ \varphi _ t\xrightarrow {d_ S}\varphi _0 \]
as $t\to 0+$.
Corollary 16.9.38B1559025EC82
Proof. First note that for each $j=0,\ldots ,n$,
\[ \lim _{t\to 0+}\int _ X \theta _{\varphi _ t}^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\varphi _0}^ j\wedge \theta _{V_{\theta }}^{n-j}. \]
So thanks to corollary 16.9.33, it remains to argue that for all $j=0,\ldots ,n$,
\[ \lim _{t\to 0+}\int _ X \theta _{\varphi _ t\lor \varphi _0}^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\varphi _0}^ j\wedge \theta _{V_{\theta }}^{n-j}. \]
Observe that for $t\in (0,1)$, we have
\[ \varphi _ t\lor \varphi _0=t(\varphi _1\lor \varphi _0)+(1-t)\varphi _0, \]
so the desired limit follows by the same binomial expansion. $\square$
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