How to hide equation number with using align

Either use \begin{align*}...\end{align*} or use \nonumber for a specific equation to be suppressed in an align environment.

This might get tedious if all equations in an align environment should be unnumbered.

General rule: An environment or command with * most times means: 'Do not number'

The same is true for alignat and alignat* environments and equation and equation* environments from amsmath, the later two are for a single equation only.

\documentclass[conference]{IEEEtran}

\usepackage{amsmath}

\begin{document}

\begin{align}
f_{n}(\beta,\lambda)&= \lambda(1-\lambda/n)^{n-1} \int_{0} ^{1} g_{n}(\beta,\lambda)d\alpha \nonumber \\
                     & \leq \lambda(1-\lambda/n)^{n-1}\int_{0}^{1} g_{n}(\beta,1)d\alpha \\
                     & \leq (1-1/n)^{n-1} \int_{0}^{1} g_{n}(\beta,1)d\alpha = f_{n}(\beta,1)
\end{align}

\end{document}

The star form \begin{align*}...\end{align*} suppresses the equation number:

\documentclass[conference]{IEEEtran}

\usepackage{amsmath}

\begin{document}

\begin{align*}
f_{n}(\beta,\lambda)&= \lambda(1-\lambda/n)^{n-1} \int_{0} ^{1}
g_{n}(\beta,\lambda)d\alpha \\
                     & \leq \lambda(1-\lambda/n)^{n-1}\int_{0}^{1}
g_{n}(\beta,1)d\alpha \\
                     & \leq (1-1/n)^{n-1} \int_{0}^{1} g_{n}(\beta,1)d\alpha
= f_{n}(\beta,1)
\end{align*}

\end{document}

This is a single equation, not a collection of equations that are being aligned, so I would use equation*, the * suppresses the equation number, and do the alignment with split, as below. This way, if I later find I need to refer to this equation and add a number all I need do is change the outer environment to equation and add a \label command.

\documentclass[conference]{IEEEtran}

\usepackage{amsmath}

\begin{document}

\begin{equation*}
  \begin{split}
    f_{n}(\beta,\lambda)
    &= \lambda(1-\lambda/n)^{n-1} \int_{0} ^{1} g_{n}(\beta,\lambda)
    \, d\alpha \\
    & \leq \lambda(1-\lambda/n)^{n-1}\int_{0}^{1} g_{n}(\beta,1)
    \, d\alpha \\
    & \leq (1-1/n)^{n-1} \int_{0}^{1} g_{n}(\beta,1) \, d\alpha
    = f_{n}(\beta,1)
  \end{split}
\end{equation*}

\end{document}

I have added a standard thin space \, before the differentials and would assume you have some punctuation to add at the end of the final line.