Why is this statement about uniform convergence to $\cos x$ false? 
There is a sequence of polynomials $P_n(x)$ which converge uniformly
  to $f(x)=\cos x$ on $\mathbb{R}$.

Given $\cos x\in C[a,\ b]$ and $\epsilon>0$, by the Weierstrass Approximation theorem, there is a polynomial $p$ such that $\|\cos x-p\|_\infty<\epsilon$. So, there is a sequence of polynomials $P_n(x)$ which converge uniformly to $f(x)=\cos x$ on $[a,\ b]$. I don't know why $[a,\ b]$ cannot be enlarged into $\mathbb{R}$.

It can't be extended to all of $\mathbb R$ because $\cos(x) $ is bounded whereas nonconstant polynomials are never bounded. As you increase the size of the interval, the same polynomials will no longer work for the larger interval because eventually the difference will exceed $\epsilon$. 
In general no sequence of polynomials  on an unbounded set of reals can converge uniformly to a nonconstant  bounded function. The theorem only applies on an interval of finite length. 
