Suppose there is a natural number $N$. Let us reverse the digits of $N$. Now let us add both the numbers e.g. if $N=98$ then add $98$ and $89$. Theorem says that if we keep doing this finite number of times then we will get a natural number that is also palindrome.
Examples:
$1.$ Consider $N=19$ Now add $19$ and $91$ to get the result as $100$ Now add $100$ and $001$ to get the result and the palindrome number $101$
$2.$ Consider $N=9$ Now add $09$ and $90$ to get the result and the palindrome number as $99$
$3.$ Consider $N=99$ then repeat the steps for a few times to get the result and the palindrome number $79497$
Is this lemma true$?$ I tried considering it for two digit numbers $10a+b$ Reversing the digits and adding them will yield $11a+11b$ which is a symmetric function in $a$ and $b$. I am not able to prove it for two digit numbers let alone a generalized case.
Any help is greatly appreciated.