How can I show that a sequence of regular polygons with $n$ sides becomes more and more like a circle as $n \to \infty$? If we construct regular polygons with larger and larger numbers of sides, they will look more and more like circles.
That is intuitively true.
I hope you will help me to express and prove it mathematically.
Here's one serious approach: Let $f_n\colon [0,2\pi]\to \Bbb R_+$ be the function whose graph, in polar coordinates, is the regular $n$-gon centered at the origin with a vertex at $(1,0)$. Then $(f_n)$ converges uniformly to a constant function mapping any angle to $1$, whose graph is a circle.

We can also look at the limit of the area, a la Archimedes, and the perimeter.
For Polygons, as number of sides increases,the length of each side is reducing

so at ∞, the length will be equal to a point.

Each Side can be assumed to be shortened to the vertex(ie a point)

Now, the polygon becomes a locus of points which are actually the vertices of the polygon.

However, each vertex is equidistant from the centre of polygon, so the resulting polygon 

becomes a locus of points, equidistant from a common centre point, which is a circle. 

After Edit: 
Now, what remains, is to prove that at n->∞, length of side tends to zero, ie a point.  

It can be proved by contradiction.
1) Suppose at n->∞, length of each side is not equal to a point, but is L.for n-> ∞, total    length = n*L = ∞.

Now, total length is infinite,however, polygon is a closed figure, so its total length is
obviously finite.

Thus this contradicts that length is infinite, which in turn contradicts our original supposition. 

Therefore, at n-> ∞, length of each side tends to 0, ie each side becomes a point.