negation of mathematical statements- Real Analysis example

You made a small but important mistake in translating this to symbols. The actual statement would be better written as $$ \forall {t_n} \in (x,b) : (t_n\rightarrow x\Rightarrow f(t_n)\rightarrow q)$$ As you can see from the parentheses I added, the quantifier is outside the implication. To negate the whole sentence, you change $\forall$ to $\exists$ then negate the implication, which results in $$ \exists {t_n} \in (x,b) : (t_n\rightarrow x\wedge f(t_n) \nrightarrow q)$$ If you’re confused about negating an implication, remember that $A\Rightarrow B$ is equivalent to $B\vee\neg A$.