Convexity of coordinate of a convex function

Question

Let $f$ be convex in $\mathbb{R}^n$. Fix $x_2,...,x_n$ and consider $g(x_1) = f(x_1,x_2,...,x_n)$. Is $g$ convex?

I think the problem should be straightfoward and tried to find to prove:

$g(\lambda x_1 + (1 - \lambda) x_1')$

$ = f(\lambda x_1 + (1 - \lambda)x_1', x_2,...,x_n)$

$ \leq \lambda f(x_1, x_2,...,x_n) + (1 - \lambda)f(x_1', x_2,...,x_n) $

$= \lambda g(x_1) + (1 - \lambda) g(x_1')$

and a counter example.

Thanks!