I'm learning about convergence of sequences in the space $\Bbb R^{\Bbb N}$ and there are couple of confusing examples. First one I have is that if $f_n \in \Bbb R^{\Bbb N}$ is given by
$$\begin{align} f_1 &= \left( \, 1 \, , 2 \, , 3 \, , \color{red}{4} \, , 5 \, , \dotsc \right) \\ f_2 &= \left( \tfrac{1}{2},\tfrac{2}{2}, \tfrac{3}{2}, \color{red}{\tfrac{4}{2}}, \tfrac{5}{2}, \dotsc \right) \\ f_3 &= \left( \tfrac{1}{3}, \tfrac{2}{3}, \tfrac{3}{3}, \color{red}{\tfrac{4}{3}}, \tfrac{5}{3}, \dotsc \right) \\ & \ \ \vdots \end{align} $$ then $f_n$ converges to the zero function. This I can see since $f_n(x)=\frac{x}{n} \to 0$.
The second one is that if I have $$f_1=(1,0,0...), f_2=(1,2,0,0,...), f_3=(1,2,3,0,0,...), \dots$$ then why does this sequence converge to $(1,2,3,4, \dots)$? and also if
$$f_1=(0,1,2,3...),f_2=(0,0,1,2,3...), f_3=(0,0,0,1,2,3,...), \dots$$ then why does this sequence converge to the zero function also?
Basic fact: $(f_n) \to f$ in $\Bbb R^{\Bbb N}$ iff $\forall m \in \Bbb N: f_n(m) \to f(m)$ in $\Bbb R$.
For the first example we have $f_n(m) = \frac{m}{n}$ and so for each fixed $m$ we have the sequence $\frac{m}{n}$ as $n \to \infty$ to consider which clearly converges to $0$ and so the limit is the constant sequence $0$.
For the second example we have
$$f_n(m) = \begin{cases} 0 & m < n\\ m & m \ge n \end{cases} $$ and so for fixed $m$ and $n \to \infty$ we just have a sequence that is constant $m$ from index $m$ onwards so in that coordinate it converges to $m$. Ergo $f(m)=m$ (the identity ) is the pointwise limit.
For the last example we have $$f_n(m) = \begin{cases} m & m < n\\ 0 & m \ge n \end{cases} $$
and so we have for fixed $m$ always a sequence that is eventually $0$ and so $f\equiv 0$ is the pointwise limit.