If ${X_n} < \infty$, then $\sup {X_n} < \infty$, True or False? This maybe very bad but please correct me. Given $X_n &gt;0$.

Suppose ${X_n} &lt; \infty$, then $\max{X_n} + 1 &lt; \infty$.

This means $\max{X_n} + 1$ is a upper bound of ${X_n}$.

Thus, $\sup {X_n} \le \max{X_n} + 1 &lt; \infty$

Why is this wrong? or right?
The notation $X_n &lt; \infty$ is a bit sloppy, because if it is read literally, it holds for a specific value $X_n$, even if the use of subscript would suggest that $X_n$ is part of a sequence of values.  If we then talk about $\sup X_n$, then the context of a sequence is made clear, but it still doesn't imply that all of the terms of the sequence are bounded above.  For that, you need to explicitly state something like $$\forall n, \quad X_n &lt; \infty,$$ or $$X_n &lt; \infty, \quad n = 1, 2, \ldots.$$  If this is the intended meaning, then the counterexample $X_n = n$ suffices, since for each $n = 1, 2, \ldots$, it is true that $n &lt; \infty$, yet $\sup n = \infty$.  The statement $$\forall n, \quad X_n &lt; \infty$$ is not equivalent to saying $$\exists C &lt; \infty \quad \text{s.t.} \quad \forall n, \quad X_n &lt; C.$$  The reason why is because if one wanted to say the second statement, one would not have written $X_n &lt; \infty$ in the first place.  One would have simply written $X_n &lt; C$.
To say that $X_n &lt; \infty$ means that there exists a (finite) number $C$ such that $$\forall\, n\ ,\quad X_{n}&lt;C\ ;$$ in other words the number $C$ is an upper bound for the sequence $X_n\ .$ Since $\sup X_n$ is the smallest upper bound for the sequence, the number $\sup X_n$ cannot be greater than the upper bound $C\ ,$ hence $$\sup X_n \leq C\ .$$