For $\epsilon>0$, let $(e_{1},\ldots,e_{r})$ be an $e^{-\epsilon}$-orthogonal basis of $V$ with respect to $\|\raisebox{1.72218pt}{.}\|$. Then, by Proposition 1.9, $(e_{1},\ldots,e_{r})$ forms an $e^{-\epsilon}$-orthogonal basis of $V_{k^{\prime}}$ and $V_{k^{\prime\prime}}$ with respect to $\|\raisebox{1.72218pt}{.}\|_{k^{\prime}}$ and $\|\raisebox{1.72218pt}{.}\|_{k^{\prime\prime}}$, respectively, so that $(e_{1},\ldots,e_{r})$ is also an $e^{-\epsilon}$-orthogonal basis of $V_{k^{\prime\prime}}$ with respect to $\|\raisebox{1.72218pt}{.}\|_{k^{\prime},k^{\prime\prime}}$. Note that $\|e_{i}\|=\|e_{i}\|_{k^{\prime\prime}}=\|e_{i}\|_{k^{\prime},k^{\prime\prime}}$ for all $i=1,\ldots,r$. Thus, for $a^{\prime\prime}_{1},\ldots,a^{\prime\prime}_{r}\in k^{\prime\prime}$,

$$\|a^{\prime\prime}_{1}e_{1}+\ldots+a^{\prime\prime}_{r}e_{r}\|_{k^{\prime},k^{\prime\prime}}\leq\max\{|a^{\prime\prime}_{1}|^{\prime\prime}\|e_{1}\|,\ldots,|a^{\prime\prime}_{r}|^{\prime\prime}\|e_{r}\|\}\\ \leq e^{\epsilon}\|a^{\prime\prime}_{1}e_{1}+\ldots+a^{\prime\prime}_{r}e_{r}\|_{k^{\prime\prime}}$$

and

$$\|a^{\prime\prime}_{1}e_{1}+\ldots+a^{\prime\prime}_{r}e_{r}\|_{k^{\prime\prime}}\leq\max\{|a^{\prime\prime}_{1}|^{\prime\prime}\|e_{1}\|,\ldots,|a^{\prime\prime}_{r}|^{\prime\prime}\|e_{r}\|\}\\ \leq e^{\epsilon}\|a^{\prime\prime}_{1}e_{1}+\ldots+a^{\prime\prime}_{r}e_{r}\|_{k^{\prime},k^{\prime\prime}}.$$

Thus, we have the assertion by taking $\epsilon\to 0$. ∎