I am trying to understand if this statement holds true:
Let $F:\mathbb{R}\to\mathbb{R}$ be a coercive and lower semicontinuous function. Let $a\in \mathbb{R}$. Thus exists $$\min_{x\in [a, +\infty)} F(x).$$
In my opinion it holds true since we are in a particular case of this more general result:
${\bf Theorem:}$ Let $H$ be an Hilbert space and $K\subset H, K\neq\emptyset$ be a closed and convex subset. If $F:H\to\mathbb{R}$ is a weakly lower semicontinuous and coercive functional. Thus $\min_{u\in K} F(u)$ exists.
In my case $H$ is a Hilbert space of dimension $1$ (so weak and strong convergence coincide) and $[a, +\infty)$ is a closed and convex subset.
Could someone please help me to understand if my reasoning holds true or am I missing something?
Thank you!