[proofplan] Expand $\|v + w\|^2$ using sesquilinearity of the inner product, bound the real part of $(v, w)$ using the Cauchy-Schwarz inequality, and take square roots. [/proofplan] [step:Expand $\|v + w\|^2$ and bound using Cauchy-Schwarz] Expanding the squared norm: \begin{align*} \|v + w\|^2 &= (v + w, v + w) = \|v\|^2 + (v, w) + \overline{(v, w)} + \|w\|^2 \\ &= \|v\|^2 + 2\,\mathrm{Re}\,(v, w) + \|w\|^2. \end{align*} Since $\mathrm{Re}\,(v, w) \leq |(v, w)| \leq \|v\|\,\|w\|$ by the [Cauchy-Schwarz Inequality](/theorems/432): \begin{align*} \|v + w\|^2 \leq \|v\|^2 + 2\|v\|\,\|w\| + \|w\|^2 = (\|v\| + \|w\|)^2. \end{align*} Taking square roots (both sides are non-negative): \begin{align*} \|v + w\| \leq \|v\| + \|w\|. \end{align*} [/step]