[proofplan] The cases $p = 1$ and $p = \infty$ follow directly from the ordinary triangle inequality. For $1 < p < \infty$, we bound $|f + g|^p$ by $(|f| + |g|)|f + g|^{p-1}$, apply the [Holder Inequality](/theorems/516) to each summand with conjugate exponent $q = p/(p-1)$, and simplify by dividing out a common factor of $\|f + g\|_p^{p/q}$. [/proofplan] [step:Bound $|f + g|^p$ using the triangle inequality] By the triangle inequality $|f + g| \leq |f| + |g|$, so \begin{align*} |f + g|^p = |f + g| \cdot |f + g|^{p-1} \leq (|f| + |g|)|f + g|^{p-1} = |f| \cdot |f + g|^{p-1} + |g| \cdot |f + g|^{p-1}. \end{align*} [/step] [step:Apply Holder's inequality to each term with conjugate exponent $q = p/(p-1)$] Let $q = p/(p-1)$ be the conjugate exponent. Since $(p-1)q = p$, the function $|f + g|^{p-1}$ belongs to $L^q$ whenever $f + g \in L^p$. Applying the [Holder Inequality](/theorems/516) to each term, \begin{align*} \int_E |f| \cdot |f + g|^{p-1} \, d\mu &\leq \|f\|_p \cdot \||f + g|^{p-1}\|_q = \|f\|_p \cdot \|f + g\|_p^{p/q}, \\ \int_E |g| \cdot |f + g|^{p-1} \, d\mu &\leq \|g\|_p \cdot \|f + g\|_p^{p/q}. \end{align*} [/step] [step:Add the bounds and divide by $\|f + g\|_p^{p/q}$ to conclude] Adding the two bounds, \begin{align*} \|f + g\|_p^p = \int_E |f + g|^p \, d\mu \leq (\|f\|_p + \|g\|_p) \cdot \|f + g\|_p^{p/q}. \end{align*} If $\|f + g\|_p = 0$ the inequality is trivial. Otherwise, dividing both sides by $\|f + g\|_p^{p/q}$ and noting that $p - p/q = p(1 - 1/q) = p/p = 1$, we obtain $\|f + g\|_p \leq \|f\|_p + \|g\|_p$. [/step]