Finding the interior of a set of continuous functions [closed]

Let $C[0,1]$ be the family of continuous functions on $[0,1]$ with thenorm $||x||_{\infty}=\max\limits_{0\leq t\leq 1}|x(t)|.$ Let$A=\{x\in C[0,1]| ||x||_{\infty}\leq 1, x(1)=1\}.$ Prove that$A$ is closed and bounded. Find $\text{Int}A$.

I already prove $A$ is closed and bounded since $A\subset B(0,2)$, $0\in C[0,1]$. I'm stuck at finding $\text{Int}A$. $x\in\text{Int}A\iff\exists r>0: B(x,r)\subset A$. How to find $\text{Int}A$.