I came across a rather unusual proof but I’m not sure if it is correct or incorrect. I want to prove that, for a sequence of functions $f_n:\mathbb{N}\rightarrow [0,1]$
$$\lim_{n\rightarrow \infty}\quad f_n(x)=f(x)$$
I found a proof by induction on $x$. Since the domain is $\mathbb{N}$ they prove that
$$\lim_{n\rightarrow \infty}\quad f_n(0)=f(0)$$ $$\lim_{n\rightarrow \infty}\quad f_n(1)=f(1)$$
And then suppose that
$$\lim_{n\rightarrow \infty}\quad f_n(k)=f(k)$$
holds. Then it is just left to prove that
$$\lim_{n\rightarrow \infty}\quad f_n(k+1)=f(k+1)$$
Can you give the arguments why this is correct or incorrect?