What's the flaw in "collapsed rounding"?

A couple of years ago, I asked my high school math teacher the following question and she couldn't give me an answer. I had since forgotten about it, but now I'm curious to find the answer again.

Let's say I have the number 1.49. Now, obviously rounding this to the nearest integer would yield the result of 1. But if I were to first round it to the nearest tenths place, I'd get 1.5, and rounding that results in 2. Even as someone who's not very well-versed in Mathematics, I can tell you that this is incorrect. My question is why? What's wrong with rounding this way. I get the practical reasons, like this is increasing the margin of error for something like taxes, but are there any mathematical reasons as to why this is incorrect?

• Successive rounding could even take you from $1.4444445$ to $2$ Apr 11, 2018 at 13:46
• Rounding gives rise to all sorts of mathematical oddities. Read about the Alabama Paradox for a serious, real world example.
– lulu
Apr 11, 2018 at 13:51
• FYI, here is a related question that has a lot more references and description of the "double rounding" problem: math.stackexchange.com/q/599772/19658 Feb 14 at 16:43

When you round to the nearest integer you guarantee that the correct value is within $\pm 0.5$ of your rounded value. You have demonstrated that collapsed rounding violates this.