Square of Small-Digit Palindromic Number is Palindromic

Theorem
Let $n$ be an integer such that the sum of the squares of the digits of $n$ in decimal representation is less than $10$.

Let $n$ be palindromic.

Then $n^2$ is also palindromic.

The sequence of such numbers begins:
 * $0, 1, 2, 3, 11, 22, 101, 111, 121, 202, 212, 1001, 1111, \dots$

Proof
Let $\displaystyle n = \sum_{k \mathop = 0}^m a_k 10^k$ be a number satisfying the conditions above.

Then:

Consider $\displaystyle n^2 = \paren {\sum_{k \mathop = 0}^m a_k 10^k}^2$.

From definition of Multiplication of Polynomials, the coefficient of $10^l$ in the product is:

so no carries occur in the multiplication, and this form satisfies Basis Representation Theorem.

Moreover:

so the coefficient of $10^{2 m - l}$ is equal to the coefficient of $10^l$ in the expansion of $n^2$.

This shows that $n^2$ is palindromic.