Show this function is not cauchy in the metric space?

Question

Let X be the set of all continuous real valued functions on [0,1]. Let $\{f_n\} = \{t^n\}$ be a sequence in X. Let $d_1(f,g) = \displaystyle{\max_{t \; \in [0,1]}}{\; |f(t)- g(t)|}$.

i) Show that $\{f_n\}$ is not convergent in the metric space (X,$d_1$).

ii) Show that $\{f_n\}$ is not Cauchy in the metric space (X,$d_1$).

My idea for convergence is to assume that $\{f_n\}$ does converge and then reach a contradiction but I do not understand how to start that. I managed to prove that $\{f_n\}$ does not converge to $0$ on (X,$d_1$) if that helps. Once I have that $\{f_n\}$ is not convergent, does ii) follow as a direct consequence from that?

Answer 1

ii). Assume that it were, then for some $N$, $|t^{n}-t^{m}|<1/2$ for all $t\in[0,1]$ and $n\geq m\geq N$. For $t\in[0,1)$, and taking $n\rightarrow\infty$, one has $|t^{N}|<1/2$ for all such $t$, now taking $t\rightarrow 1^{-}$ to get a contradiction.

Note that ii) implies i).

Answer 2

ii follows from I since $(X,d_1)$ is (known to be) complete...

For the argument see https://en.m.wikipedia.org/wiki/Complete_metric_space...