Could you please help me solve this question relating to sequences?
Suppose that a sequence $\{a_n\}$ converges to $\pi$. Then the sequence $\{\cos(a_n)\}$ _____.
The answer is converges to $-1$ but why is that case? I thought since $\cos(\pi) = 0$, the sequence should converge to 0?
Thanks! :)
$\cos$ is continuous on $\Bbb R$, so $$\lim_{n\to\infty} \cos(a_n)=\cos(\lim_{n\to\infty} a_n)=\color{red}{\cos(\pi)=-1}.$$
This is due to the continuity of $\cos(x)$. Generally a function $f$ is continuous if $$ \lim_{x \to a} f(x) = f(a) $$ which in qualitative terms says if I move $x$ closer and closer to $a$ then $f(x)$ will get closer and closer to $f(a)$ (this getting closer and closer occurs at about the same rate).
Now in some simplistic terms we can consider a sequence $\{a_n\}$ to converge to a point $\pi$ to mean that as $n$ increases unboundedly, the terms of the sequence get closer and closer to $\pi$.
Intuitively, then, we can make the jump that since $a_n$ gets closer and closer to $\pi$ we can say that $$ \lim_{n \to \infty} f(a_n) = f\left( \lim_{n \to \infty} a_n \right) $$
Of course this can all be proven rigorously (which I'll gladly show you how if you want) but I hope this provides some intuition.
Also note that $\cos \pi = -1$ not $0$ ;D
