Let $\mu_1,\mu_2$ be two probability measures on a discrete set $X$. Define the total variation distance between $\mu_1,\mu_2$ in terms of coupling as $$d(\mu_1,\mu_2) = \inf_{\gamma \in \Gamma(\mu_1,\mu_2)}\gamma(\{(x_1,x_2) | x_1\neq x_2\})$$ where $\Gamma(\mu_1,\mu_2)$ is the set of couplings of $\mu_1,\mu_2$, a coupling being a probability measure on $X\times X$ with marginals $\mu_1$ and $\mu_2$.
I want to prove that if $\inf_{x} \mu_1(x) \geq \epsilon$ and $\inf_{x}\mu_2(x) \geq \epsilon$, then $$d(\mu_1,\mu_2) \leq 1-\epsilon$$ Is the following correct?
Consider the product coupling $\gamma(x_1,x_2) = \mu_1(x_1)\mu_2(x_2)$. Then \begin{align*} d(\mu_1,\mu_2) &\leq \gamma(\{ (x_1,x_2) | x_1\neq x_2 \}) \\ &= 1 - \gamma(\{(x_1,x_2) | x_1 = x_2\}) \\ &= 1- \sum\limits_{x}\gamma(x,x) \\ &= 1 - \sum\limits_{x}\mu_1(x)\mu_2(x) \\ &\leq 1- \sum\limits_{x}\mu_1(x)\epsilon \\ &= 1- \epsilon \end{align*}