# Deciding whether two metrics are topologically equivalent in the space $C^1()$

Consider the space $C^1([0,1])$ and the function $d:C([0,1])\times C([0,1]) \to \mathbb R$ defined as $d(f,g)=|f(0)-g(0)|+sup_{x \in [0,1]}|f'(x)-g'(x)|$. Decide whether the metrics $d$ and $d_{\infty}$ are topologically equivalent in $C^1([0,1])$ (where $d_{\infty}=sup_{x \in [0,1]}|f(x)-g(x)|)$

My attempt at a solution:

If two metrics are topologically equivalent, then they have the same convergent sequences. Honestly, I couldn’t do anything. I am trying to define a sequence of functions $\{f_n\}_{n \in \mathbb N}$ such that $f_n \to f$ in, for instance, $(C^1([0,1]),d)$ but $f_n \not \to f$ in $(C^1([0,1]),d_{\infty})$. Could it be this two metrics are topologically equivalent? If this is the case, how could I prove it? If not, I would appreciate any hint to find an adequate sequence of functions that works for what I am trying to prove.

#### Solutions Collecting From Web of "Deciding whether two metrics are topologically equivalent in the space $C^1()$"

Set $f_n(x) = \frac{1}{n} \sin(nx)$. It converges to zero in $(C^1([0,1]),d_{\infty})$ but it doesn’t converge to zero in $(C^1([0,1]),d)$, because there is always a point $x$ such that $f’_n(x) = 1$.