$\left(f_{n}\right)_{n\in\mathbb{N}}$ where $$ f_n\left(x\right) = \begin{cases} n, & \text{if } x\geq n \\ 1, & \text{ if } x < n \end{cases} $$
I need to determine whether the above is pointwise convergent. I know that this involves finding the limit but I am not sure how to go about approaching 2 cases.

  • $\begingroup$ Your first sentence is not a sentence. ${}\qquad{}$ $\endgroup$ – Michael Hardy Jan 9 '16 at 22:08
  • 1
    $\begingroup$ Fix an $x_0$. Look whether $\bigl(f_n(x_0)\bigr)_{n\in\mathbb{N}}$ converges. $\endgroup$ – Daniel Fischer Jan 9 '16 at 22:09

You don't actually have to deal with two cases here. Pick any $y \in \mathbb{R}$. Let $N > y$, then $\forall n > N$, $f_n(y) = 1$.

This tells us that $(f_n)$ converges pointwise to $1$ at $y$ for all $y \in \mathbb{R}$


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