Proving $B=\{f:f\in C_{[0,1]}\text{ and }\:d(f,0)\leqslant 1\}$ is not compact

Question

Let $(C_{[0,1]},d)$ be the metric space defined with the supremum metric. Let $B=\{f:f\in C_{[0,1]}\text{ and}\:d(f,0)\leqslant 1\}$ where $0$ denotes the constant function form $[0,1]$ into $\mathbb{R}$.

Prove that $B$ is not compact.

Since we are working on a metric space I can use the compactness definition that every convergent sequence has a convergent subsequence so I am trying to find a function that acts as a counterexample to prove the space is not compact.

Question:

Is my strategy right? If so, how should I find the sequence of functions?

Thanks in advance!

Answer 1

You can indeed use a sequence, e.g. $f_n(x) = x^n$ is a good candidate.