If for any sequence $t_n \to t$ from the left we have $\lim_{n} y(t_n)=l$, then do we have $\lim_{s\to t^-}y(s)=l$?

Question

Let $(E,r)$ be a metric space. If for any sequence $t_n \to t$ from the left we have $\lim_{n} y(t_n)=l$, then do we have $\lim_{s\to t^-}y(s)=l$?

I used this result to prove that if for each $t>0$ both of these limits exist, then $y^-(t)=y(t-)$ for all $t>0$. Here we define for $F$ a dense subset of $[0,\infty)$, and $x:F\to E$,

$$y(t)\equiv\lim_{s\to t+, s\in F} x(s)$$exists, which is right continuous and $$y^- (t)\equiv \lim_{s\to t-, s\in F} x(s)$$ which is left continuous on $(0,\infty)$.