Let $f : [0,1] \rightarrow \mathbb{R}$ be continuous. Suppose that $(x_n)_{x \in \mathbb{N}}$ is a sequence in $[0,1]$, with $f(x_n) \rightarrow 0$. Show that there exists $\psi \in [0,1]$ with $f(\psi) = 0$
Do we show that $\psi$ can be found within the sequence. And if the sequence tends to zero, then we know that all sub-sequences tend to the same value?