Suppose the objective function to be minimized is $$F(\theta) = \|y - X \theta\|_2^2 + \sum_{i=1}^p \lambda_i |\theta_i|$$ where $\theta$ is the independent variable which is feasible in $\mathbb{R}^p$. Other quantities $y \in \mathbb{R}^{n}$, $X \in\mathbb{R}^{n \times p}$, individual penalties $\lambda_i \geq 0$ are all fixed. Note that some $\lambda_i$'s might be zero. We focus on high dimension so $p>n$.
$F(\theta)$ is convex. But how to show that there exists $\widehat{\theta} \in \mathbb{R}^p$ such that for any $\theta \in \mathbb{R}^p$, $F(\theta) \geq F(\widehat{\theta})$?
You'll need to the following:
An extended value function $f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$ is said to be coercive if $f(x)\to\infty$ as $\|x\|\to\infty$.
A classic result in convex analysis states that a proper lower semi-continuous and coercive function must be bounded below and have (global) minimizers.
With the above, show that the LASSO objective is coercive by showing the function $\theta\mapsto\|y-X\theta\|^2$ is so. Hint: Expand the square.