# Is \( \infty + 1 > \infty \)?

This is part of a series on common misconceptions.

Is this true or false?

\[ \infty + 1 > \infty \]

**Why some people say it's true:** Adding one to any number always makes it bigger.

**Why some people say it's false:** Infinity is not a real number, so this arithmetic is not valid. Also, according to the infinity rules, \(x + \infty = \infty\), with \(x\) is any real number. Hence, \(\infty + 1 = \infty\)

The statement is \(\color{red}{\textbf{false}}\).

Proof:We cannot do ANY ARITHMETIC OPERATION with infinity, since it is a concept, not a number.

But let's take a look at the limit to prove it is wrong.

Let \[\lim_{x\to\infty}{f(x)} = \infty\] and \[\lim_{x\to\infty}{g(x)} = 1\]

We have: \[\lim_{x\to\infty}{f(x)} + \lim_{x\to\infty}{g(x)} = \lim_{x\to\infty}{(f(x)+g(x))} = \infty + 1 = \infty\]

Hence, the statement is \(\color{red}{\textbf{false}}\).

Rebuttal: But what if you have a very large number, say, \(x = 3141592653589\dots\), will it be larger infinity?

Reply: You may want to see Is \( \infty + \infty > \infty \) ?

Rebuttal: If you cannot do any arithmetic operation with infinity, what does \(\infty + 1 = \infty\) mean in your proof?

Reply:That means you use the one of the properties of infinity to evaluate the limit of a function.

**See Also**

**Cite as:**Is \( \infty + 1 > \infty \)?.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/is-infty-1-infty/