Reading through Hartshorne's Algebraic Geometry, and reading different notes about cohomology of sheaves, I have often seen the argument that if you have an exact sequence of coherent sheaves over a projective scheme $X$ $$0\to \mathcal{F}'\to\mathcal{F}\to\mathcal{F}''\to 0$$ then $$0\to\mathcal{F}'(n)\to\mathcal{F}(n)\to\mathcal{F}''(n)\to 0$$ is again an exact sequence.
Is this true always? And why is this true?
Tensoring by an invertible object (in this case $\mathcal{O}(n)$) is an equivalence of categories, and equivalences of categories preserve all limits and colimits, and in particular preserve exact sequences.