Number of connected components in a even graph

Let $G$ be an even graph. Show that $c(G-v)\leq \frac{d(v)}{2}$, $\forall v \in V(G)$.

Where $c(G)$ is the number of connected components of $G$ and $d(v)$ is the degree of $v$.

If $G$ is even then there is an Euler tour in $G$. Hence, any edge in $G$ must be contained in a cycle, and because of that cannot be cut edge.

I know that $G$ still may hold cut vertexes, but what can I say about them ? I know that since $v$ is in an Euler tour then there are $2$ edges inciding in $v$ for each cycle that contains it. So, $G-v$ has $2d(v)$ less edges than $G$, but only $d(v)$ connected components. I.e. you have to remove two edges before disconnecting a cycle. Is that a proof?

You don't say it, but it is required that $G$ is connected.
The Euler circuit is the key to your proof. Take an example of an Euler circuit in $G$ starting at $v$, and this trail revisits $v$ exactly $d(v)/2$ times (including the terminal visit). Thus removing $v$ breaks that trail into $d(v)/2$ trails, each of which must be confined to a connected component.