Let $c$ be the set of all convergent real sequences $\mathbb N \to \mathbb R$ and $c_0$ be the subspace of all the sequences that converges to zero. We can consider $c$ as a Banach space under the norm $||\cdot||_{\infty}$, $c_0$ it's also a Banach space because it's a closed subspace of the complete space $c$.
I want to find a projection $P: c\to c_0$ (i.e linear, continuous and such that $P^2=P$) and prove that this projection has norm always greater than 1.
We are under the assumption that $c_0\subset c \subset l^{\infty}$ ($l^{\infty}$ denotes the bounded sequences endowed with the $||\cdot||_{\infty}$ norm.
$c$ is the space of convergent sequences (it said this in the title, and I just edited your question to include it). Thus it makes sense to define $P:c\to c_0$ by $P(x_0,x_1,x_2,\ldots)=(x_0-\lim\limits_{n\to\infty}x_n,x_1-\lim\limits_{n\to\infty}x_n,x_2-\lim\limits_{n\to\infty}x_n,\ldots)$.
Regardless of choice, if $P^2=P$ and $P(c)=c_0$, then there exists $x\in c$ such that $Px\neq 0$, and $P(Px)=Px$ implies that $\|P\|\geq 1$.