This may be a trivial question for people dealing with infinite series a lot - but I am not and feel a bit insecure here.
Suppose I have a function $f:\mathbb R\to\mathbb R; t\mapsto f(t)$. I am not sure if it matters, but we can consider it to be smooth ($\mathcal C^\infty$), and I am only interested in it on an interval $[t_*,\infty)$.
Now consider the infinite sequence $(t_i)_{i=0}^\infty$ with $t_i:=t_*+i\epsilon$ for some $\epsilon>0$. This then induces the infinite sequence $(f_i)_{i=0}^\infty$ with $f_i:=f(t_i)$.
I now want to take the continuum limit, i.e. I want to state that for arbitrarily small steps between the $t_i$, the series $(f_i)$ approaches the continuous function $f(t)$.
Intuitively that is clear to me, since it is in some sense by construction. What makes me cautious is, that if I take the limit $\epsilon\to0$, then $t_i\to t_*$ and $f_i\to f(t_*)$ for all $i$.
How do I get this straight?
This can happen when limits are taken in different orders.
The two limits are $\epsilon \to 0$ and $n \to \infty$ (the number of points).
What you have shown is that taking the limits in different orders produces different results. This is not surprising.