Suppose that the function $f:[0,1]\rightarrow \mathbb{R}$ is continuous and that $f\left(x\right)>2$ if $0\leq x<1$. Is it necessarily true that $f\left(1\right)>2$?
My attempt:
Yes, using the sequence definition of continuous.
Since $f$ is continuous at $x=1$, so we can take $x_n\rightarrow 1$ as $n\rightarrow \infty$, then $f\left(x_n\right) \rightarrow f\left(1\right)$ as $n\rightarrow \infty$.
Furthermore, $f\left(x_n\right)>2$ $\forall$ $x_n$ where $0\leq x_n < 1$.
$\implies f\left(1\right)>2$.
Does this make sense?