Define what it means for a set of vectors in a vector space $V$ to be linearly dependent. Prove from the definition that any set of $n+1$ vectors in $\mathbb{R}^{n}$ is linearly dependent.

Using this or otherwise, prove that if $V$ has a finite basis consisting of $n$ elements, then any basis of $V$ has exactly $n$ elements.

Let $V$ be the vector space of bounded continuous functions on $\mathbb{R}$. Show that $V$ is infinite dimensional.