[proofplan] We exploit the triangle inequality twice — once with $x = (x - y) + y$ and once with the roles of $x$ and $y$ swapped — to obtain two one-sided bounds on the difference $|x| - |y|$. Combining these two bounds yields the desired two-sided inequality $\bigl||x| - |y|\bigr| \leq |x - y|$. [/proofplan] [step:Apply the triangle inequality to $x = (x - y) + y$ to bound $|x| - |y|$ from above] Write $x = (x - y) + y$. The triangle inequality for the absolute value $|\cdot|$ on $K$ gives \begin{align*} |x| = |(x - y) + y| \leq |x - y| + |y|. \end{align*} Subtracting $|y|$ from both sides yields \begin{align*} |x| - |y| \leq |x - y|. \tag{$*$} \end{align*} [guided] The idea is simple: the triangle inequality tells us that $|x|$ cannot exceed $|x - y| + |y|$, which rearranges to a bound on the signed difference $|x| - |y|$. Write $x = (x - y) + y$ and apply the triangle inequality for the absolute value $|\cdot|$ on $K$: \begin{align*} |x| = |(x - y) + y| \leq |x - y| + |y|. \end{align*} Subtracting $|y|$ from both sides: \begin{align*} |x| - |y| \leq |x - y|. \tag{$*$} \end{align*} This gives a one-sided bound. To turn it into a two-sided bound (with the outer standard absolute value on $\mathbb{R}$), we need the symmetric inequality $|y| - |x| \leq |x - y|$ as well, which we obtain in the next step. [/guided] [/step] [step:Swap $x$ and $y$ to bound $|y| - |x|$ from above] Interchanging $x$ and $y$ in inequality $(*)$ gives \begin{align*} |y| - |x| \leq |y - x|. \end{align*} Since $|y - x| = |(-1)(x - y)| = |{-1}| \cdot |x - y| = |x - y|$ (using multiplicativity and $|{-1}| = 1$), we obtain \begin{align*} |y| - |x| \leq |x - y|, \end{align*} or equivalently, \begin{align*} -(|x| - |y|) \leq |x - y|. \tag{$**$} \end{align*} [/step] [step:Combine both bounds to conclude the reverse triangle inequality] Inequalities $(*)$ and $(**)$ together state that \begin{align*} -(|x - y|) \leq |x| - |y| \leq |x - y|. \end{align*} By the characterisation of the standard absolute value on $\mathbb{R}$ (namely $|t| \leq M$ if and only if $-M \leq t \leq M$ for $t \in \mathbb{R}$ and $M \geq 0$), this is equivalent to \begin{align*} \bigl| |x| - |y| \bigr| \leq |x - y|, \end{align*} where the outer absolute value is the standard absolute value on $\mathbb{R}$. This is the desired inequality. [/step]