TITLE: Is there anything wrong with this proposed proof of the irrationality of Euler's constant?
QUESTION [5 upvotes]: Let $\{\lambda_n\}$ be the sequence given by $H_n - \ln n$. We claim that $\lambda_n$ is irrational for every integer $n>1$ and justify this by the following argument:
Assume that $\lambda_k$ is rational for some integer $k>1$ such that $H_k - \ln k = p/q$ where $p$ and $q$ are integers.
Rearranging the above we arrive at $H_k - p/q = \ln k$, which implies that $\ln k$ is rational since $H_k$ is rational. But we know that $\ln k$ is irrational for all integers $k>1$, hence we reach a contradiction.
Therefore, $\lambda_n$ is irrational for all integers $n>1$. Hence the limit as $n$ tends to infinity is irrational, and we are done.

REPLY [20 votes]: For example,
$$
\lambda_n := e - \sum_{k=0}^n\frac{1}{k!}
$$
are all irrational numbers, but their limit is zero.

REPLY [10 votes]: You are implicitly claiming that the set of irrational numbers is closed in $\mathbb R$, which it is not. Indeed its closure is $\mathbb R$ (it is a dense subset), i.e. every real number is the limit of a sequence of irrational numbers.